Proof techniques

Analysis proofs reuse a small set of moves. This page groups the proofs in §1–§18 by the move they lean on, so you can see one idea at work in many places, or find a worked model when you need to write a similar proof yourself. Citations such as Theorem 9.3 are links: hover over or focus one to see the statement, then open its proof on that page.

A proof often uses several moves. Each is listed under the one that does the most work.

Proof by contradiction

Assume the conclusion fails and derive something impossible. Reach for it when the statement says “no such object exists” or when negating it hands you something concrete to work with.

• Theorem 4.6 (Archimedean Property): if some multiple $na$ never exceeds $b$, the supremum of $\{na\}$ contradicts itself.
• Corollary 2.3 and its use on $\sqrt2$ in §2: a rational root would have to be an integer dividing $c_0$, and none is.
• Non-convergence in §8, Example 4: if $(-1)^n \to a$, then $a$ is at distance at least $1$ from one of $\pm1$.
• Theorem 16.3 (two decimal expansions): assuming two expansions of $x$ differ in an unexpected way forces a contradiction.
• Theorem 18.1 (boundedness on $[a,b]$): an unbounded continuous $f$ gives $|f(x_n)| > n$, and Bolzano–Weierstrass turns that into a contradiction at the limit point. See also Subsequence extraction below.

The ε/2 trick

Split a target error $\varepsilon$ into two halves, one for each of two quantities you can control, and combine them with the triangle inequality.

• Uniqueness of limits in §7: both $|s_n - s|$ and $|s_n - t|$ are made smaller than $\varepsilon/2$.
• Theorem 9.3 (sums of limits).
• Theorem 9.4 (products of limits): the same idea with $\varepsilon/(2M)$ and $\varepsilon/(2(|t|+1))$ so each term stays below $\varepsilon/2$.
• Lemma 10.9 (convergent sequences are Cauchy).

Adding and subtracting a middle term

Insert a term and subtract it again so that each difference is one you already control.

• Theorem 9.4: $s_n t_n - st = s_n(t_n - t) + t(s_n - s)$.
• Theorem 17.3 (continuity of $|f|$): $\big||f(x_n)| - |f(x_0)|\big| \le |f(x_n) - f(x_0)|$.
• Corollary 3.6 (distance triangle inequality): apply Theorem 3.5(iii) to $(a-b) + (b-c)$.
• Theorem 14.6 (Comparison Test): the triangle inequality bounds a tail of $\sum b_n$ by the same tail of $\sum a_n$.

Bounding by a constant

Use a finite maximum to get one bound that works for every term.

• Theorem 9.1 (convergent sequences are bounded): take $\max\{|s|+1, |s_1|, \dots, |s_N|\}$.
• Lemma 10.10 (Cauchy sequences are bounded): the same recipe.
• Theorem 9.4 uses Theorem 9.1 so that $|s_n|$ can be replaced by $M$.

Squeezing and comparison

Trap a hard quantity between two easy ones, or dominate it by something whose behaviour you know.

• The squeeze lemma, Exercise 8.5(a), and its uses in Theorem 9.7(d), where $1 \le a^{1/n} \le n^{1/n}$.
• Theorem 9.7(b): write $|a| = \frac1{1+b}$ and bound $|a|^n$ with the binomial theorem.
• Theorem 14.6 (Comparison Test).
• Theorem 14.9 (Root Test): for large $n$, $|a_n| < (\alpha+\varepsilon)^n$, compared with a geometric series.
• Theorem 15.1 ($p$-series): compare partial sums with $\int_1^n x^{-p}\,dx$ in both directions.
• Theorem 12.2 and Corollary 12.3: the ratio and root quantities are ordered, so a limit of the ratios pins down the limit of the roots.

Supremum and infimum arguments

Name the least upper bound, then use that anything smaller is not an upper bound.

• Corollary 4.5 (inf exists): reflect $S$ to $-S$ and apply the Completeness Axiom.
• Theorem 10.2 (bounded monotone sequences converge): $u - \varepsilon$ is not an upper bound, so some $s_N > u - \varepsilon$.
• Theorem 18.2 (Intermediate Value Theorem): take $x_0 = \sup\{x : f(x) < y\}$ and approach it from inside the set.
• Theorem 10.7 and Theorem 10.11: $\liminf$ and $\limsup$ are limits of infima and suprema of tails.

Bisection and nested intervals

Halve a set repeatedly, keep the half that still has the property you need, and let the nested pieces shrink to a point.

• Theorem 11.5 (Bolzano–Weierstrass), halving proof described in §11, with the nested interval property from §4.
• Theorem 13.10 (nested closed bounded sets are nonempty intersection) and Proposition 13.13 ($k$-cells are compact): halve a cell into $2^k$ pieces, keep one that has no finite subcover, then use Theorem 13.10.
• §18 also has a bisection widget that homes in on the crossing in Theorem 18.2, although the proof there uses a supremum.

Subsequence extraction

Build a subsequence with a useful property, then pass to its limit.

• Theorem 11.4 (every sequence has a monotonic subsequence): pick the “dominant” terms, or else a rising run.
• Theorem 11.5 (Bolzano–Weierstrass): Theorem 11.4 plus Theorem 10.2.
• Theorem 11.7 and Theorem 11.8: the subsequential limits of $\limsup$ and $\liminf$.
• Theorem 12.1: a subsequence along which $t_n \to \limsup t_n$.
• Theorem 13.5 (Bolzano–Weierstrass in $\mathbb{R}^k$): extract coordinate by coordinate, as in Theorem 13.4.
• Theorem 13.10 and Theorem 18.1: a convergent subsequence of chosen points.

The sequential characterisation

Translate a statement about functions or sets into one about sequences, where the limit theorems of §9 apply.

• Theorem 17.2 (ε–δ continuity is equivalent to sequential continuity).
• Theorem 17.3, Theorem 17.4 and Theorem 17.5 (algebra and composition of continuous functions): each proof takes a sequence $x_n \to x_0$ and applies the matching limit theorem.
• Lemma 13.3 and Theorem 13.4: convergence and Cauchy-ness in $\mathbb{R}^k$ reduce to each coordinate.

Cases and exhaustion

Split into cases that cover everything, and prove each.

• Theorem 3.5 (properties of $|a|$): four sign cases for $|ab| = |a||b|$.
• Theorem 11.4: infinitely many dominant terms, or only finitely many.
• Theorem 12.1: $\limsup t_n$ finite or infinite.
• Theorem 18.6 (injective continuous functions are monotone): consider where $f(b)$ sits relative to $f(a)$ and $f(c)$, and use the Intermediate Value Theorem on each side.

Induction

Prove a base case and a step, then conclude for every $n$.

• §1: the principle itself and its first uses.
• The recursive sequences in §10, where $s_n > 0$ is proved by induction before monotonicity is checked.
• The geometric-series exercises in §14, where $a_n = a_1 r^{n-1}$ comes from induction.

Algebra from axioms

Unwind a definition and apply the axioms in order. No cleverness, only careful bookkeeping.

• Theorem 3.1 (field consequences) and Theorem 3.2 (ordered field consequences).
• Theorem 3.5 (absolute value).
• Corollary 14.5 and Corollary 14.7 (immediate consequences of the Cauchy criterion and the Comparison Test).

Constructive and explicit proofs

Build the object, then check it works.

• Theorem 4.7 (denseness of $\mathbb{Q}$): choose $n$ with $n(b-a) > 1$, then an integer $m$ between $an$ and $bn$.
• Theorem 16.2 (existence of decimal expansions): define the digits with the floor function, one at a time.
• Theorem 16.5 (rational iff repeating): both directions build the expansion or the fraction explicitly.
• Theorem 9.7(a): choose $N = (1/\varepsilon)^{1/p}$.
• Theorem 9.9 and Theorem 9.10: choose $N$ from $M$, working backward from the goal.

Practise a technique

These proofs are interactive: put the steps in order, fill the gaps, or find the flawed line.

• Contradiction: Archimedean Property (Theorem 4.6), in order, in §4.
• ε/2 and the triangle inequality: limit of a sum (Theorem 9.3), fill the gaps, in §9.
• Supremum arguments: Monotone Convergence Theorem (Theorem 10.2) in §10, and the Intermediate Value Theorem (Theorem 18.2) in §18.
• Comparison: Ratio Test (Theorem 14.8), fill the gaps, in §14.

For the mistakes that trip people up most often when using these moves, see Common mistakes.