Exercises

Prove $1^2 + 2^2 + \cdots + n^2 = \tfrac16 n(n+1)(2n+1)$ for all positive integers $n$ .

Prove $3 + 11 + \cdots + (8n - 5) = 4n^2 - n$ for all positive integers $n$ .

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ for all positive integers $n$ .

Exercise 1.4Warm-upInductionComputation

Guess a formula for $1 + 3 + \cdots + (2n-1)$ by evaluating the sum for $n = 1, 2, 3, 4$ . Prove your formula using mathematical induction.

Exercise 1.5Warm-upInduction

Prove $1 + \tfrac12 + \tfrac14 + \cdots + \tfrac{1}{2^n} = 2 - \tfrac{1}{2^n}$ for all positive integers $n$ .

Exercise 1.6StandardInduction

Prove $11^n - 4^n$ is divisible by $7$ when $n$ is a positive integer.

Exercise 1.7StandardInduction

Prove $7^n - 6n - 1$ is divisible by $36$ for all positive integers $n$ .

Exercise 1.8StandardInductionInequalities

The principle of induction extends as follows: $P_m, P_{m+1}, \dots$ are all true provided (i) $P_m$ is true and (ii) $P_{n+1}$ is true whenever $P_n$ is true and $n \ge m$ . Prove …

Exercise 1.9StandardInductionInequalities

Decide for which integers the inequality $2^n > n^2$ is true. Prove your claim in (a) by mathematical induction.

Exercise 1.10StandardInduction

Prove $(2n+1) + (2n+3) + (2n+5) + \cdots + (4n-1) = 3n^2$ for all positive integers $n$ .

Exercise 1.11Warm-upInductionComputation

For each $n \in \mathbb{N}$ , let $P_n$ denote the assertion " $n^2 + 5n + 1$ is an even integer." Prove $P_{n+1}$ is true whenever $P_n$ is true. For which $n$ is $P_n$ …

Exercise 1.12ChallengeInduction

For $n \in \mathbb{N}$ let $n! = 1 \cdot 2 \cdots n$ , let $0! = 1$ , and define $\binom{n}{k} = \dfrac{n!}{k!\,(n-k)!}$ for $k = 0, 1, \dots, n$ . The binomial theorem asserts …

Exercise 2.1StandardContradiction

Show $\sqrt3$ , $\sqrt5$ , $\sqrt7$ , $\sqrt{24}$ and $\sqrt{31}$ are not rational numbers.

Exercise 2.2StandardContradiction

Show $\sqrt[3]{2}$ , $\sqrt[7]{5}$ and $\sqrt[4]{13}$ are not rational numbers.

Exercise 2.3StandardContradiction

Show $\sqrt{2 + \sqrt2}$ is not a rational number.

Exercise 2.4StandardContradiction

Show $\sqrt[3]{5 - \sqrt3}$ is not a rational number.

Exercise 2.5StandardContradiction

Show $\left[3 + \sqrt2\right]^{2/3}$ is not a rational number.

Exercise 2.6Warm-upExplain whyComputation

In connection with Example 6, discuss why $4 - 7b^2$ is rational if $b$ is rational.

Exercise 2.7Warm-upComputation

Show the following irrational-looking expressions are actually rational numbers. $\sqrt{4 + 2\sqrt3} - \sqrt3$ $\sqrt{6 + 4\sqrt2} - \sqrt2$

Exercise 2.8StandardComputationContradiction

Find all rational solutions of the equation $x^8 - 4x^5 + 13x^3 - 7x + 1 = 0$ .

Exercise 3.1Warm-upFrom the definitionsExplain why

Which of the properties A1–A4, M1–M4, DL, O1–O5 fail for $\mathbb{N}$ ? Which of these properties fail for $\mathbb{Z}$ ?

Exercise 3.2Warm-upFrom the definitions

The commutative law A2 was used in the proof of (ii) in Theorem 3.1. Where? The commutative law A2 was also used in the proof of (iii) in Theorem 3.1. Where?

Exercise 3.3StandardFrom the definitions

Prove (iv) and (v) of Theorem 3.1.

Exercise 3.4StandardFrom the definitions

Prove (v) and (vii) of Theorem 3.2.

Exercise 3.5StandardInequalities

Show $|b| \le a$ if and only if $-a \le b \le a$ . Prove $\bigl||a| - |b|\bigr| \le |a - b|$ for all $a, b \in \mathbb{R}$ .

Exercise 3.6StandardInequalitiesInduction

Prove $|a + b + c| \le |a| + |b| + |c|$ for all $a, b, c \in \mathbb{R}$ . (Apply the triangle inequality twice. Do not consider eight cases.) Use induction to prove …

Exercise 3.7Warm-upInequalities

Show $|b| < a$ if and only if $-a < b < a$ . Show $|a - b| < c$ if and only if $b - c < a < b + c$ . Show $|a - b| \le c$ if and only if $b - c \le a \le b + c$ .

Exercise 3.8StandardInequalitiesContradiction

Let $a, b \in \mathbb{R}$ . Show if $a \le b_1$ for every $b_1 > b$ , then $a \le b$ .

Exercise 4.1Warm-upComputationSup & inf

For each set below that is bounded above, list three upper bounds for the set. Otherwise write "NOT BOUNDED ABOVE" or "NBA." (An integer $p \ge 2$ …

Exercise 4.2Warm-upComputationSup & inf

Repeat Exercise 4.1 for lower bounds.

Exercise 4.3Warm-upComputationSup & inf

For each set in Exercise 4.1, give its supremum if it has one. Otherwise write "NO sup."

Exercise 4.4Warm-upComputationSup & inf

Repeat Exercise 4.3 for infima (the plural of infimum).

Exercise 4.5Warm-upSup & inf

Let $S$ be a nonempty subset of $\mathbb{R}$ that is bounded above. Prove that if $\sup S$ belongs to $S$ , then $\sup S = \max S$ . (Your proof should be very short.)

Exercise 4.6StandardSup & inf

Let $S$ be a nonempty bounded subset of $\mathbb{R}$ . Prove $\inf S \le \sup S$ . (This is almost obvious; your proof should be short.) What can you say about $S$ if …

Exercise 4.7StandardSup & inf

Let $S$ and $T$ be nonempty bounded subsets of $\mathbb{R}$ . Prove that if $S \subseteq T$ , then $\inf T \le \inf S \le \sup S \le \sup T$ . Prove …

Exercise 4.8StandardSup & inf

Let $S$ and $T$ be nonempty subsets of $\mathbb{R}$ such that $s \le t$ for all $s \in S$ and $t \in T$ . Observe $S$ is bounded above and $T$ is bounded below. Prove $\sup S \le \inf T$ …

Exercise 4.9StandardSup & inf

Complete the proof that $\inf S = -\sup(-S)$ in Corollary 4.5 by proving (1) and (2).

Exercise 4.10StandardSup & inf

Prove that if $a > 0$ , then there exists $n \in \mathbb{N}$ such that $\frac1n < a < n$ .

Exercise 4.11StandardSup & inf

Consider $a, b \in \mathbb{R}$ with $a < b$ . Use Denseness of $\mathbb{Q}$ 4.7 to show there are infinitely many rationals between $a$ and $b$ .

Exercise 4.12StandardSup & infContradiction

Let $\mathbb{I}$ be the set of real numbers that are not rational; these are the irrational numbers. Prove that if $a < b$ , then there exists $x \in \mathbb{I}$ with $a < x < b$ .

Exercise 4.13Warm-upInequalities

Prove the following are equivalent for real numbers $a, b, c$ (either all of them hold or none does): $|a - b| < c$ , $b - c < a < b + c$ , $a \in (b - c, b + c)$ .

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Exercise 4.14ChallengeSup & inf

Let $A$ and $B$ be nonempty bounded subsets of $\mathbb{R}$ , and let $A + B = \{a + b : a \in A,\ b \in B\}$ . Prove $\sup(A + B) = \sup A + \sup B$ . Prove …

Exercise 4.15StandardInequalitiesContradiction

Let $a, b \in \mathbb{R}$ . Show that if $a \le b + \frac1n$ for all $n \in \mathbb{N}$ , then $a \le b$ . Compare Exercise 3.8.

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Exercise 4.16StandardSup & inf

Show $\sup\{r \in \mathbb{Q} : r < a\} = a$ for each $a \in \mathbb{R}$ .

Exercise 5.1Warm-upComputation

Write the following sets in interval notation: $\{x \in \mathbb{R} : x < 0\}$ $\{x \in \mathbb{R} : x^3 \le 8\}$ $\{x^2 : x \in \mathbb{R}\}$ $\{x \in \mathbb{R} : x^2 < 8\}$

Exercise 5.2Warm-upSup & infComputation

Give the infimum and supremum of each set listed in Exercise 5.1.

Exercise 5.3Warm-upSup & infComputation

Give the infimum and supremum of each unbounded set listed in Exercise 4.1.

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Exercise 5.4StandardSup & inf

Let $S$ be a nonempty subset of $\mathbb{R}$ , and let $-S = \{-s : s \in S\}$ . Prove $\inf S = -\sup(-S)$ .

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Exercise 5.5Warm-upSup & inf

Prove $\inf S \le \sup S$ for every nonempty subset $S$ of $\mathbb{R}$ . Compare Exercise 4.6(a).

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Exercise 5.6StandardSup & inf

Let $S$ and $T$ be nonempty subsets of $\mathbb{R}$ with $S \subseteq T$ . Prove $\inf T \le \inf S \le \sup S \le \sup T$ . Compare Exercise 4.7(a).

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Exercise 5.7StandardSup & inf

Finish Example 1 by verifying the equality involving infimums.

Exercise 6.1StandardFrom the definitions

Consider $s, t \in \mathbb{Q}$ . Show $s \le t$ if and only if $s^* \subseteq t^*$ ; $s = t$ if and only if $s^* = t^*$ ; $(s + t)^* = s^* + t^*$ . (Note that $s^* + t^*$ …

Exercise 6.2ChallengeFrom the definitions

Show that if $\alpha$ and $\beta$ are Dedekind cuts, then so is $\alpha + \beta = \{r_1 + r_2 : r_1 \in \alpha \text{ and } r_2 \in \beta\}$ .

Exercise 6.3ChallengeFrom the definitions

Show $\alpha + 0^* = \alpha$ for all Dedekind cuts $\alpha$ . We claimed, without proof, that addition of Dedekind cuts satisfies A4: for each cut $\alpha$ there is a cut …

Exercise 6.4ChallengeFrom the definitionsExamples & counterexamples

Let $\alpha$ and $\beta$ be Dedekind cuts and define the "product" $\alpha \cdot \beta = \{r_1 r_2 : r_1 \in \alpha \text{ and } r_2 \in \beta\}$ …

Exercise 6.5ChallengeFrom the definitionsExamples & counterexamples

Show $\{r \in \mathbb{Q} : r^3 < 2\}$ is a Dedekind cut, but $\{r \in \mathbb{Q} : r^2 < 2\}$ is not. Does the Dedekind cut $\{r \in \mathbb{Q} : r^3 < 2\}$ …

Exercise 7.1Warm-upComputation

Write out the first five terms of the following sequences. $s_n = \dfrac{1}{3n+1}$ $b_n = \dfrac{3n+1}{4n-1}$ $c_n = \dfrac{n}{3^n}$ $\sin\!\left(\dfrac{n\pi}{4}\right)$

Exercise 7.2Warm-upComputation

For each sequence in Exercise 7.1, determine whether it converges. If it converges, give its limit. No proofs are required.

Exercise 7.3Warm-upComputation

For each sequence below, determine whether it converges and, if it converges, give its limit. No proofs are required. $a_n = \dfrac{n}{n+1}$ $b_n = \dfrac{n^2+3}{n^2-3}$ …

Exercise 7.4Warm-upExamples & counterexamples

Give examples of a sequence $(x_n)$ of irrational numbers whose limit is a rational number; a sequence $(r_n)$ of rational numbers whose limit is an irrational number.

Exercise 7.5StandardComputation

Determine the following limits. No proofs are required, but show any relevant algebra. $\lim s_n$ where $s_n = \sqrt{n^2+1} - n$ $\lim\left(\sqrt{n^2+n} - n\right)$ …

Exercise 8.1Standardε–N / ε–δ

Prove the following: $\lim \dfrac{(-1)^n}{n} = 0$ $\lim \dfrac{1}{n^{1/3}} = 0$ $\lim \dfrac{2n-1}{3n+2} = \dfrac23$ $\lim \dfrac{n+6}{n^2-6} = 0$

Exercise 8.2Standardε–N / ε–δ

Determine the limits of the following sequences, and then prove your claims. $a_n = \dfrac{n}{n^2+1}$ $b_n = \dfrac{7n-19}{3n+7}$ $c_n = \dfrac{4n+3}{7n-5}$ …

Exercise 8.3Standardε–N / ε–δ

Let $(s_n)$ be a sequence of nonnegative real numbers with $\lim s_n = 0$ . Prove $\lim \sqrt{s_n} = 0$ . This completes the proof of Example 5.

Exercise 8.4Standardε–N / ε–δ

Let $(t_n)$ be a bounded sequence, i.e. there is $M$ with $|t_n| \le M$ for all $n$ , and let $(s_n)$ be a sequence with $\lim s_n = 0$ . Prove $\lim (s_n t_n) = 0$ .

Exercise 8.5Standardε–N / ε–δ

This exercise is referred to in several places. Consider sequences $(a_n)$ , $(b_n)$ , $(s_n)$ with $a_n \le s_n \le b_n$ for all $n$ and $\lim a_n = \lim b_n = s$ . Prove $\lim s_n = s$ …

Exercise 8.6Standardε–N / ε–δ

Let $(s_n)$ be a sequence in $\mathbb{R}$ . Prove $\lim s_n = 0$ if and only if $\lim |s_n| = 0$ . Observe that if $s_n = (-1)^n$ , then $\lim |s_n|$ exists, but $\lim s_n$ does not.

Exercise 8.7Standardε–N / ε–δContradiction

Show the following sequences do not converge. $\cos\left(\dfrac{n\pi}{3}\right)$ $s_n = (-1)^n n$ $\sin\left(\dfrac{n\pi}{3}\right)$

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Exercise 8.8Standardε–N / ε–δComputation

Prove the following (see Exercise 7.5): $\lim\left[\sqrt{n^2+1} - n\right] = 0$ $\lim\left[\sqrt{n^2+n} - n\right] = \frac12$ $\lim\left[\sqrt{4n^2+n} - 2n\right] = \frac14$

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Exercise 8.9Standardε–N / ε–δContradiction

Let $(s_n)$ be a sequence that converges. (This exercise is referred to in several places.) Show that if $s_n \ge a$ for all but finitely many $n$ , then $\lim s_n \ge a$ …

Exercise 8.10Standardε–N / ε–δ

Let $(s_n)$ be a convergent sequence with $\lim s_n > a$ . Prove there is a number $N$ such that $n > N$ implies $s_n > a$ .

Exercise 9.1Warm-upLimit laws

Using the limit Theorems 9.2–9.7, prove the following. Justify all steps. $\lim \dfrac{n+1}{n} = 1$ $\lim \dfrac{3n+7}{6n-5} = \dfrac12$ …

Exercise 9.2Warm-upLimit laws

Suppose $\lim x_n = 3$ , $\lim y_n = 7$ and all $y_n$ are nonzero. Determine: $\lim(x_n + y_n)$ $\lim \dfrac{3y_n - x_n}{y_n^2}$

Exercise 9.3StandardLimit laws

Suppose $\lim a_n = a$ , $\lim b_n = b$ , and $s_n = \dfrac{a_n^3 + 4a_n}{b_n^2 + 1}$ . Prove $\lim s_n = \dfrac{a^3 + 4a}{b^2 + 1}$ carefully, using the limit theorems.

Exercise 9.4StandardLimit lawsComputation

Let $s_1 = 1$ and $s_{n+1} = \sqrt{s_n + 1}$ for $n \ge 1$ . List the first four terms of $(s_n)$ . It turns out that $(s_n)$ converges. Assume this and prove the limit is …

Exercise 9.5Warm-upLimit lawsComputation

Let $t_1 = 1$ and $t_{n+1} = \dfrac{t_n^2 + 2}{2t_n}$ for $n \ge 1$ . Assume $(t_n)$ converges and find the limit.

Exercise 9.6StandardLimit lawsExplain why

Let $x_1 = 1$ and $x_{n+1} = 3x_n^2$ for $n \ge 1$ . Show that if $a = \lim x_n$ , then $a = \frac13$ or $a = 0$ . Does $\lim x_n$ …

Exercise 9.7Standardε–N / ε–δ

Complete the proof of Theorem 9.7(c), i.e. give the standard argument needed to show $\lim s_n = 0$ .

Exercise 9.8Warm-upComputation

Give the following when they exist. Otherwise assert "NOT EXIST." $\lim n^3$ $\lim(-n^3)$ $\lim(-n)^n$ $\lim(1.01)^n$ $\lim n^n$

Exercise 9.9Standardε–N / ε–δ

Suppose there is $N_0$ such that $s_n \le t_n$ for all $n > N_0$ . Prove that if $\lim s_n = +\infty$ , then $\lim t_n = +\infty$ . Prove that if $\lim t_n = -\infty$ , then …

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Exercise 9.10Standardε–N / ε–δ

Show that if $\lim s_n = +\infty$ and $k > 0$ , then $\lim(ks_n) = +\infty$ . Show $\lim s_n = +\infty$ if and only if $\lim(-s_n) = -\infty$ . Show that if $\lim s_n = +\infty$ and …

Exercise 9.11Standardε–N / ε–δ

Show that if $\lim s_n = +\infty$ and $\inf\{t_n : n \in \mathbb{N}\} > -\infty$ , then $\lim(s_n + t_n) = +\infty$ . Show that if $\lim s_n = +\infty$ and $\lim t_n > -\infty$ …

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Exercise 9.12Challengeε–N / ε–δInequalities

Assume all $s_n \ne 0$ and that the limit $L = \lim \left|\dfrac{s_{n+1}}{s_n}\right|$ exists. (This exercise is referred to in several places.) Show that if $L 1$ , then …

Exercise 9.13StandardLimit laws

Show $\lim_{n\to\infty} a^n = \begin{cases} 0 & \text{if } |a| 1 \\ \text{does not exist} & \text{if } a \le -1. \end{cases}$

Exercise 9.14StandardLimit laws

Let $p > 0$ . Use Exercise 9.12 to show …

Exercise 9.15StandardLimit laws

Show $\lim_{n\to\infty} \dfrac{a^n}{n!} = 0$ for all $a \in \mathbb{R}$ .

Exercise 9.16StandardLimit laws

Use Theorems 9.9 and 9.10 or Exercises 9.9–9.15 to prove the following: $\lim \dfrac{n^4 + 8n}{n^2 + 9} = +\infty$ $\lim\left[\dfrac{2^n}{n^2} + (-1)^n\right] = +\infty$ …

Exercise 9.17Warm-upε–N / ε–δ

Give a formal proof that $\lim n^2 = +\infty$ using only Definition 9.8.

Exercise 9.18StandardLimit lawsComputation

Verify $1 + a + a^2 + \cdots + a^n = \dfrac{1 - a^{n+1}}{1 - a}$ for $a \ne 1$ . Find $\lim_{n\to\infty}(1 + a + a^2 + \cdots + a^n)$ for $|a| < 1$ . Calculate …

Exercise 10.1Warm-upComputation

Which of the following sequences are increasing? decreasing? bounded? $\frac1n$ $\frac{(-1)^n}{n^2}$ $n^5$ $\sin\frac{n\pi}7$ $(-2)^n$ $\frac{n}{3^n}$

Exercise 10.2StandardMonotone & Cauchy

Prove Theorem 10.2 for bounded decreasing sequences.

Exercise 10.3StandardMonotone & Cauchy

For a decimal expansion $K.d_1d_2d_3\cdots$ let $(s_n)$ be as in 10.3. Prove $s_n < K + 1$ for all $n$ .

Exercise 10.4Warm-upExplain why

Discuss why Theorems 10.2 and 10.11 would fail if we restricted our world of numbers to the set $\mathbb{Q}$ of rational numbers.

Exercise 10.5StandardMonotone & Cauchy

Prove Theorem 10.4(ii).

Exercise 10.6StandardMonotone & Cauchy

Let $(s_n)$ be a sequence with $|s_{n+1} - s_n| < 2^{-n}$ for all $n$ . Prove $(s_n)$ is a Cauchy sequence and hence convergent. Is the result in (a) true if we only assume …

Exercise 10.7StandardSup & infSequences & continuity

Let $S$ be a bounded nonempty subset of $\mathbb{R}$ such that $\sup S$ is not in $S$ . Prove there is a sequence $(s_n)$ of points in $S$ with $\lim s_n = \sup S$ …

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Exercise 10.8StandardMonotone & CauchyInequalities

Let $(s_n)$ be an increasing sequence of positive numbers and define $\sigma_n = \frac1n(s_1 + s_2 + \cdots + s_n)$ . Prove $(\sigma_n)$ is increasing.

Exercise 10.9StandardMonotone & CauchyInduction

Let $s_1 = 1$ and $s_{n+1} = \left(\frac{n}{n+1}\right) s_n^2$ for $n \ge 1$ . Find $s_2$ , $s_3$ and $s_4$ . Show $\lim s_n$ exists. Prove $\lim s_n = 0$ .

Exercise 10.10StandardMonotone & CauchyInduction

Let $s_1 = 1$ and $s_{n+1} = \frac13(s_n + 1)$ for $n \ge 1$ . Find $s_2$ , $s_3$ and $s_4$ . Use induction to show $s_n > \frac12$ for all $n$ . Show $(s_n)$ is a decreasing sequence. Show …

Exercise 10.11StandardMonotone & Cauchy

Let $t_1 = 1$ and $t_{n+1} = \left[1 - \frac1{4n^2}\right] t_n$ for $n \ge 1$ . Show $\lim t_n$ exists. What do you think $\lim t_n$ is?

Exercise 10.12StandardMonotone & CauchyInduction

Let $t_1 = 1$ and $t_{n+1} = \left[1 - \frac1{(n+1)^2}\right] t_n$ for $n \ge 1$ . Show $\lim t_n$ exists. What do you think $\lim t_n$ is? Use induction to show …

Exercise 11.1Warm-upSubsequences

Let $a_n = 3 + 2(-1)^n$ for $n \in \mathbb{N}$ . List the first eight terms of $(a_n)$ …

Exercise 11.2Warm-upSubsequences

Consider $a_n = (-1)^n$ , $b_n = \frac1n$ , $c_n = n^2$ , $d_n = \frac{6n+4}{7n-3}$ …

Exercise 11.3Warm-upSubsequences

Repeat Exercise 11.2 for $s_n = \cos\frac{n\pi}3$ , $t_n = \frac{3}{4n+1}$ , $u_n = \left(-\frac12\right)^n$ , $v_n = (-1)^n + \frac1n$ .

Exercise 11.4Warm-upSubsequences

Repeat Exercise 11.2 for $w_n = (-2)^n$ , $x_n = 5^{(-1)^n}$ , $y_n = 1 + (-1)^n$ , $z_n = n\cos\frac{n\pi}4$ .

Exercise 11.5StandardSubsequences

Let $(q_n)$ be an enumeration of all the rationals in the interval $(0, 1]$ . Give the set of subsequential limits for $(q_n)$ . Give the values of $\limsup q_n$ and $\liminf q_n$ .

Exercise 11.6Warm-upSubsequencesFrom the definitions

Show every subsequence of a subsequence of a given sequence is itself a subsequence of the given sequence.

Exercise 11.7StandardSubsequences

Let $(r_n)$ be an enumeration of the set $\mathbb{Q}$ of all rational numbers. Show there exists a subsequence $(r_{n_k})$ such that $\lim_{k\to\infty} r_{n_k} = +\infty$ .

Exercise 11.8StandardSubsequencesSup & inf

Use Definition 10.6 and Exercise 5.4 to prove $\liminf s_n = -\limsup(-s_n)$ for every sequence $(s_n)$ . (This exercise is referred to in several places.)

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Exercise 11.9StandardOpen, closed, compactSubsequences

Show the closed interval $[a, b]$ is a closed set. Is there a sequence $(s_n)$ such that $(0, 1)$ is its set of subsequential limits?

Exercise 11.10StandardSubsequences

Form an infinite array whose every row is $1, \frac12, \frac13, \frac14, \frac15, \dots$ , and let $(s_n)$ …

Exercise 11.11StandardSup & infSubsequences

Let $S$ be a bounded set. Prove there is an increasing sequence $(s_n)$ of points in $S$ with $\lim s_n = \sup S$ . Compare Exercise 10.7. (If $\sup S \in S$ , it suffices to let …

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Exercise 12.1Standardlim sup & lim inf

Let $(s_n)$ and $(t_n)$ be sequences with $s_n \le t_n$ for all $n > N_0$ . Show $\liminf s_n \le \liminf t_n$ and $\limsup s_n \le \limsup t_n$ .

Builds on §9, §10

Exercise 12.2Standardlim sup & lim inf

Prove $\limsup |s_n| = 0$ if and only if $\lim s_n = 0$ .

Builds on §8, §10

Exercise 12.3Warm-uplim sup & lim infComputation

Let $(s_n)$ and $(t_n)$ repeat in cycles of four: $(s_n) = (0, 1, 2, 1,\ 0, 1, 2, 1,\ \dots), \qquad (t_n) = (2, 1, 1, 0,\ 2, 1, 1, 0,\ \dots).$ Find $\liminf s_n + \liminf t_n$ …

Exercise 12.4Standardlim sup & lim inf

Show $\limsup(s_n + t_n) \le \limsup s_n + \limsup t_n$ for bounded sequences $(s_n)$ and $(t_n)$ .

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Exercise 12.5Standardlim sup & lim inf

Use Exercises 11.8 and 12.4 to prove $\liminf(s_n + t_n) \ge \liminf s_n + \liminf t_n$ for bounded sequences $(s_n)$ and $(t_n)$ .

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Exercise 12.6Standardlim sup & lim inf

Let $(s_n)$ be a bounded sequence and $k \ge 0$ . Prove $\limsup(ks_n) = k \cdot \limsup s_n$ . Do the same for $\liminf$ . (Use Exercise 11.8.) What happens in (a) and (b) if $k < 0$ …

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Exercise 12.7Warm-uplim sup & lim inf

Prove that if $\limsup s_n = +\infty$ and $k > 0$ , then $\limsup(ks_n) = +\infty$ .

Exercise 12.8Challengelim sup & lim inf

Let $(s_n)$ and $(t_n)$ be bounded sequences of nonnegative numbers. Prove $\limsup s_n t_n \le (\limsup s_n)(\limsup t_n)$ .

Exercise 12.9Standardlim sup & lim inf

Prove that if $\lim s_n = +\infty$ and $\liminf t_n > 0$ , then $\lim s_n t_n = +\infty$ . Prove that if $\limsup s_n = +\infty$ and $\liminf t_n > 0$ , then $\limsup s_n t_n = +\infty$ …

Exercise 12.10Standardlim sup & lim inf

Prove $(s_n)$ is bounded if and only if $\limsup |s_n| < +\infty$ .

Exercise 12.11Standardlim sup & lim inf

Prove the first inequality in Theorem 12.2.

Exercise 12.12Challengelim sup & lim inf

Let $(s_n)$ be a sequence of nonnegative numbers, and let $\sigma_n = \frac1n(s_1 + s_2 + \cdots + s_n)$ . Show …

Exercise 12.13Challengelim sup & lim infSup & inf

Let $(s_n)$ be a bounded sequence in $\mathbb{R}$ . Let $A$ be the set of $a \in \mathbb{R}$ such that $\{n : s_n b\}$ is finite. Prove $\sup A = \liminf s_n$ and $\inf B = \limsup s_n$ …

Exercise 12.14Challengelim sup & lim infComputation

Calculate $\lim (n!)^{1/n}$ $\lim \frac1n (n!)^{1/n}$

Exercise 13.1StandardOpen, closed, compact

For $x, y \in \mathbb{R}^k$ let $d_1(x, y) = \max\{|x_j - y_j| : j = 1, \dots, k\}$ and $d_2(x, y) = \sum_{j=1}^k |x_j - y_j|$ . Show $d_1$ and $d_2$ are metrics for $\mathbb{R}^k$ …

Exercise 13.2StandardOpen, closed, compact

Prove (1) in Lemma 13.3. Prove the first assertion in Lemma 13.3.

Exercise 13.3StandardOpen, closed, compact

Let $B$ be the set of all bounded sequences $x = (x_1, x_2, \dots)$ and define $d(x, y) = \sup\{|x_j - y_j| : j = 1, 2, \dots\}$ . Show $d$ is a metric for $B$ . Does …

Exercise 13.4StandardOpen, closed, compact

Prove (iii) and (iv) in Discussion 13.7.

Exercise 13.5StandardOpen, closed, compact

Verify one of De Morgan's laws: $\bigcap\{S \setminus U : U \in \mathcal{U}\} = S \setminus \bigcup\{U : U \in \mathcal{U}\}$ …

Exercise 13.6StandardOpen, closed, compact

Prove Proposition 13.9.

Exercise 13.7ChallengeOpen, closed, compact

Show that every open set in $\mathbb{R}$ is the disjoint union of a finite or infinite sequence of open intervals.

Exercise 13.8Warm-upOpen, closed, compact

Verify the assertions in the first paragraph of Example 3. Verify the assertions in Example 4.

Exercise 13.9Warm-upOpen, closed, compactComputation

Find the closures of the following sets: $\{\frac1n : n \in \mathbb{N}\}$ $\mathbb{Q}$ $\{r \in \mathbb{Q} : r^2 < 2\}$

Exercise 13.10Warm-upOpen, closed, compact

Show that the interior of each of the following sets is empty: $\{\frac1n : n \in \mathbb{N}\}$ $\mathbb{Q}$ The Cantor set in Example 5

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Exercise 13.11ChallengeOpen, closed, compactSubsequences

Let $E \subseteq \mathbb{R}^k$ . Show that $E$ is compact if and only if every sequence in $E$ has a subsequence converging to a point in $E$ .

Exercise 13.12StandardOpen, closed, compact

Let $(S, d)$ be any metric space. Show that if $E$ is a closed subset of a compact set $F$ , then $E$ is compact. Show that a finite union of compact sets in $S$ is compact.

Exercise 13.13StandardOpen, closed, compactSup & inf

Let $E$ be a compact nonempty subset of $\mathbb{R}$ . Show $\sup E$ and $\inf E$ belong to $E$ .

Exercise 13.14StandardOpen, closed, compactSubsequences

Let $E$ be a compact nonempty subset of $\mathbb{R}^k$ and $\delta = \sup\{d(x, y) : x, y \in E\}$ . Show $E$ contains points $x_0, y_0$ with $d(x_0, y_0) = \delta$ .

Exercise 13.15ChallengeOpen, closed, compact

Let $(B, d)$ be as in Exercise 13.3, and let $F$ consist of all $x \in B$ with $\sup\{|x_j|\} \le 1$ . Show $F$ is closed and bounded. (A set $F$ in a metric space is bounded if …

Exercise 14.1Warm-upSeries tests

Determine which of the following series converge. Justify your answers. $\sum \dfrac{n^4}{2^n}$ $\sum \dfrac{2^n}{n!}$ $\sum \dfrac{n^2}{3^n}$ $\sum \dfrac{n!}{n^4 + 3}$ …

Exercise 14.2Warm-upSeries tests

Repeat Exercise 14.1 for the following. $\sum \dfrac{n-1}{n^2}$ $\sum (-1)^n$ $\sum \dfrac{3n}{n^3}$ $\sum \dfrac{n^3}{3^n}$ $\sum \dfrac{n^2}{n!}$ $\sum \dfrac1{n^n}$ …

Exercise 14.3Warm-upSeries tests

Repeat Exercise 14.1 for the following. $\sum \dfrac1{\sqrt{n!}}$ $\sum \dfrac{2 + \cos n}{3^n}$ $\sum \dfrac1{2^n + n}$ $\sum \left(\tfrac12\right)^n (50 + n^2)$ …

Exercise 14.4StandardSeries tests

Repeat Exercise 14.1 for the following. $\sum_{n=2}^\infty \dfrac1{[n + (-1)^n]^2}$ $\sum \left[\sqrt{n+1} - \sqrt n\right]$ $\sum \dfrac{n!}{n^n}$

Exercise 14.5Warm-upSeries testsLimit laws

Suppose $\sum a_n = A$ and $\sum b_n = B$ with $A, B$ real. Use limit theorems from §9 to quickly prove: $\sum(a_n + b_n) = A + B$ . $\sum ka_n = kA$ for $k \in \mathbb{R}$ . Is …

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Exercise 14.6StandardSeries tests

Prove that if $\sum |a_n|$ converges and $(b_n)$ is bounded, then $\sum a_n b_n$ converges. Observe that Corollary 14.7 is a special case of part (a).

Exercise 14.7StandardSeries tests

Prove that if $\sum a_n$ is a convergent series of nonnegative numbers and $p > 1$ , then $\sum a_n^p$ converges.

Exercise 14.8StandardSeries tests

Show that if $\sum a_n$ and $\sum b_n$ are convergent series of nonnegative numbers, then $\sum \sqrt{a_n b_n}$ converges.

Exercise 14.9StandardSeries tests

The convergence of a series does not depend on any finite number of its terms, though its value does. Precisely: if $\{n \in \mathbb{N} : a_n \ne b_n\}$ is finite, then …

Exercise 14.10StandardSeries testsExamples & counterexamples

Find a series $\sum a_n$ that diverges by the Root Test but for which the Ratio Test gives no information. Compare Example 8.

Exercise 14.11Warm-upSeries tests

Let $(a_n)$ be a sequence of nonzero reals such that the ratios $\left(\frac{a_{n+1}}{a_n}\right)$ form a constant sequence. Show $\sum a_n$ is a geometric series.

Exercise 14.12ChallengeSeries testsSubsequences

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence with $\liminf |a_n| = 0$ . Prove there is a subsequence $(a_{n_k})$ such that $\sum_{k=1}^\infty a_{n_k}$ converges.

Builds on §11

Exercise 14.13Warm-upSeries testsComputation

It is often much harder to find the value of an infinite sum than to show it exists. Here are some sums that can be handled. Calculate …

Exercise 14.14StandardSeries tests

Prove $\sum_{n=1}^\infty \frac1n$ diverges by comparing with $\sum_{n=2}^\infty a_n$ , where …

Exercise 15.1Warm-upSeries tests

Determine which of the following series converge. Justify your answers. $\sum \dfrac{(-1)^n}{n}$ $\sum \dfrac{(-1)^n n!}{2^n}$

Exercise 15.2StandardSeries tests

Repeat Exercise 15.1 for the following. $\sum \left[\sin\left(\tfrac{n\pi}6\right)\right]^n$ $\sum \left[\sin\left(\tfrac{n\pi}7\right)\right]^n$

Exercise 15.3StandardSeries tests

Show $\displaystyle\sum_{n=2}^\infty \frac1{n(\log n)^p}$ converges if and only if $p > 1$ .

Exercise 15.4StandardSeries tests

Determine which of the following series converge. Justify your answers. $\sum_{n=2}^\infty \dfrac1{\sqrt n \log n}$ $\sum_{n=2}^\infty \dfrac{\log n}{n}$ …

Exercise 15.5Warm-upExplain why

Why didn't we use the Comparison Test to prove Theorem 15.1 for $p > 1$ ?

Exercise 15.6StandardSeries testsExamples & counterexamples

Give an example of a divergent series $\sum a_n$ for which $\sum a_n^2$ converges. Observe that if $\sum a_n$ is a convergent series of nonnegative terms, then $\sum a_n^2$ …

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Exercise 15.7ChallengeSeries testsMonotone & Cauchy

Prove that if $(a_n)$ is a decreasing sequence of real numbers and $\sum a_n$ converges, then $\lim n a_n = 0$ . Use (a) to give another proof that $\sum \frac1n$ diverges.

Exercise 15.8ChallengeSeries tests

Formulate and prove a general integral test as advised in 15.2.

Exercise 16.1Warm-upComputation

Show $2.74\overline{9}$ and $2.75\overline{0}$ are both decimal expansions for $\frac{11}4$ . Which of these expansions arises from the long division process in 16.1?

Exercise 16.2StandardFrom the definitions

Verify the claims in the first paragraph of the proof of Theorem 16.3.

Exercise 16.3StandardSeries tests

Suppose $\sum a_n$ and $\sum b_n$ are convergent series of nonnegative numbers. Show that if $a_n \le b_n$ for all $n$ and $a_n < b_n$ for at least one $n$ , then $\sum a_n < \sum b_n$ .

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Exercise 16.4Warm-upComputation

Write the following repeating decimals as rationals, i.e. as fractions of integers. $.\overline{2}$ $.\overline{02}$ $.0\overline{2}$ $3.\overline{14}$ $.\overline{10}$ …

Exercise 16.5Warm-upComputation

Find the decimal expansions of the following rational numbers. $\frac18$ $\frac1{16}$ $\frac23$ $\frac79$ $\frac6{11}$ $\frac{22}7$

Exercise 16.6Warm-upComputation

Find the decimal expansions of $\frac17, \frac27, \frac37, \frac47, \frac57$ and $\frac67$ . Note the interesting pattern.

Exercise 16.7StandardExplain why

Is $.1234567891011121314151617181920212223242526\cdots$ rational?

Exercise 16.8ChallengeContradiction

Let $(s_n)$ be a sequence of numbers in $(0, 1)$ . Each $s_n$ has a decimal expansion $0.d_1^{(n)} d_2^{(n)} d_3^{(n)} \cdots$ . For each $n$ , let $e_n = 6$ if $d_n^{(n)} \ne 6$ and …

Exercise 16.9ChallengeMonotone & CauchySeries tests

Let $\gamma_n = \left(\sum_{k=1}^n \frac1k\right) - \log_e n = \sum_{k=1}^n \frac1k - \int_1^n \frac1t\,dt$ . Show $(\gamma_n)$ is a decreasing sequence. Show …

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Exercise 16.10Warm-upContradiction

In Example 6(b) we showed $\pi^2$ is irrational. Use this to show $\pi$ is irrational. What can you say about $\sqrt\pi$ and $\sqrt[3]\pi$ ? $\pi^4$ ?

Exercise 17.1Warm-upComputation

Let $f(x) = \sqrt{4 - x}$ for $x \le 4$ and $g(x) = x^2$ for all $x \in \mathbb{R}$ . Give the domains of $f + g$ , $fg$ , $f \circ g$ and $g \circ f$ . Find $f \circ g(0)$ , $g \circ f(0)$ , …

Exercise 17.2Warm-upComputation

Let $f(x) = 4$ for $x \ge 0$ , $f(x) = 0$ for $x < 0$ , and $g(x) = x^2$ for all $x$ . Thus $\operatorname{dom}(f) = \operatorname{dom}(g) = \mathbb{R}$ . Determine $f + g$ , $fg$ , $f \circ g$ , …

Exercise 17.3Warm-upSequences & continuity

Accept on faith that the following are continuous on their domains: $\sin x$ , $\cos x$ , $e^x$ , $2^x$ , $\log_e x$ for $x > 0$ , and $x^p$ for $x > 0$ ( $p$ …

Exercise 17.4Standardε–N / ε–δ

Prove the function $\sqrt x$ is continuous on its domain $[0, \infty)$ .

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Exercise 17.5StandardSequences & continuityInduction

Prove that if $m \in \mathbb{N}$ , then $f(x) = x^m$ is continuous on $\mathbb{R}$ . Prove every polynomial $p(x) = a_0 + a_1 x + \cdots + a_n x^n$ is continuous on $\mathbb{R}$ .

Exercise 17.6StandardSequences & continuity

A rational function is a function $f = p/q$ with $p, q$ polynomials; its domain is $\{x \in \mathbb{R} : q(x) \ne 0\}$ . Prove every rational function is continuous.

Exercise 17.7Warm-upSequences & continuity

Observe that if $k \in \mathbb{R}$ , then $g(x) = kx$ is continuous by Exercise 17.5. Prove $|x|$ is a continuous function on $\mathbb{R}$ …

Exercise 17.8StandardSequences & continuity

Let $f$ and $g$ be real-valued functions. Show $\min(f, g) = \frac12(f + g) - \frac12|f - g|$ . Show $\min(f, g) = -\max(-f, -g)$ . Use (a) or (b) to prove that if $f$ and $g$ …

Exercise 17.9Standardε–N / ε–δ

Prove each function is continuous at $x_0$ by verifying the ε–δ property of Theorem 17.2. (Try (a)–(c) in the widget above.) $f(x) = x^2$ , $x_0 = 2$ ; $f(x) = \sqrt x$ , $x_0 = 0$ ; …

Exercise 17.10Standardε–N / ε–δExamples & counterexamples

Prove the following functions are discontinuous at the indicated points, using either Definition 17.1 or the ε–δ property. $f(x) = 1$ for $x > 0$ and $f(x) = 0$ for $x \le 0$ , …

Exercise 17.11ChallengeSequences & continuitySubsequences

Let $f$ be real-valued with $\operatorname{dom}(f) \subseteq \mathbb{R}$ . Prove $f$ is continuous at $x_0$ if and only if, for every *monotonic* sequence $(x_n)$ in …

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Exercise 17.12StandardSequences & continuity

Let $f$ be a continuous real-valued function with domain $(a, b)$ . Show that if $f(r) = 0$ for each rational $r$ in $(a, b)$ , then $f(x) = 0$ for all $x \in (a, b)$ . Let $f$ and $g$ …

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Exercise 17.13StandardSequences & continuityExamples & counterexamples

Let $f(x) = 1$ for rational $x$ and $f(x) = 0$ for irrational $x$ . Show $f$ is discontinuous at every $x$ in $\mathbb{R}$ . Let $h(x) = x$ for rational $x$ and $h(x) = 0$ for irrational $x$ …

Exercise 17.14ChallengeExamples & counterexamplesε–N / ε–δ

For each nonzero rational $x$ , write $x = \frac pq$ with $p, q$ integers having no common factors and $q > 0$ , and define $f(x) = \frac1q$ . Also define $f(0) = 1$ and $f(x) = 0$ …

Exercise 17.15StandardSequences & continuity

Let $f$ be real-valued with $\operatorname{dom}(f) \subseteq \mathbb{R}$ . Show $f$ is continuous at $x_0$ in $\operatorname{dom}(f)$ if and only if, for every sequence $(x_n)$ in …

Exercise 17.16Warm-upComputation

The postage-stamp function $P$ is defined by $P(x) = A$ for $0 \le x < 1$ and $P(x) = A + Bn$ for $n \le x < n + 1$ . So $P$ takes the value $A$ on $[0, 1)$ , $A + B$ on $[1, 2)$ , $A + 2B$ on …

Exercise 18.1Warm-upIVT & EVT

Let $f$ be as in Theorem 18.1. Show that if $-f$ assumes its maximum at $x_0 \in [a, b]$ , then $f$ assumes its minimum at $x_0$ .

Exercise 18.2Warm-upExplain why

Reread the proof of Theorem 18.1 with $[a, b]$ replaced by $(a, b)$ . Where does it break down? Discuss.

Exercise 18.3Warm-upComputation

Use calculus to find the maximum and minimum of $f(x) = x^3 - 6x^2 + 9x + 1$ on $[0, 5)$ .

Exercise 18.4StandardExamples & counterexamplesSubsequences

Let $S \subseteq \mathbb{R}$ and suppose there is a sequence $(x_n)$ in $S$ converging to a number $x_0 \notin S$ . Show there exists an unbounded continuous function on $S$ .

Exercise 18.5StandardIVT & EVT

Let $f$ and $g$ be continuous on $[a, b]$ with $f(a) \ge g(a)$ and $f(b) \le g(b)$ . Prove $f(x_0) = g(x_0)$ for at least one $x_0$ in $[a, b]$ …

Exercise 18.6Warm-upIVT & EVT

Prove $x = \cos x$ for some $x$ in $(0, \frac\pi2)$ . (Find it numerically in the widget above.)

Exercise 18.7Warm-upIVT & EVT

Prove $x e^x = 2$ for some $x$ in $(0, 1)$ .

Exercise 18.8Warm-upIVT & EVT

Suppose $f$ is a real-valued continuous function on $\mathbb{R}$ and $f(a)f(b) < 0$ for some $a, b \in \mathbb{R}$ . Prove there exists $x$ between $a$ and $b$ such that $f(x) = 0$ .

Exercise 18.9StandardIVT & EVT

Prove that a polynomial function $f$ of odd degree has at least one real root.

Exercise 18.10ChallengeIVT & EVT

Suppose $f$ is continuous on $[0, 2]$ and $f(0) = f(2)$ . Prove there exist $x, y$ in $[0, 2]$ such that $|y - x| = 1$ and $f(x) = f(y)$ .

Exercise 18.11StandardIVT & EVT

Show that if $f$ is strictly increasing on an interval $I$ , then $-f$ is strictly decreasing on $I$ . State and prove Theorems 18.4 and 18.5 for strictly decreasing functions.

Exercise 18.12ChallengeIVT & EVTSequences & continuity

Let $f(x) = \sin\left(\frac1x\right)$ for $x \ne 0$ and $f(0) = 0$ . Observe that $f$ is discontinuous at $0$ by Exercise 17.10(b). Show $f$ has the intermediate value property on …

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