TheoremStatement: AI-adaptedProof: Literature-sourcedprecheck passverified 2026-08-05 (gpt-5.6-sol-codex-subscription)

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The squeeze theorem

Statement

Let $(a_{k})$ , $(x_{k})$ and $(b_{k})$ be sequences of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences) with

$a_{k} \leq x_{k} \leq b_{k} eventually,$

and suppose $(a_{k})$ and $(b_{k})$ both converge to the same real $L$ (Limits and Cauchy sequences of reals). Then $(x_{k})$ converges to $L$ .

The middle sequence is not assumed to converge; that is the point of the theorem, and it is why the squeeze is a tool for establishing convergence rather than for computing a limit already known to exist.

Facts & Assumptions

Given: Sequences $(a_{k})$ , $(x_{k})$ , $(b_{k})$ of reals, an index $K_{0} \in N$ with $a_{k} \leq x_{k} \leq b_{k}$ for every $k \geq K_{0}$ , and a real $L$ such that $(a_{k})$ converges to $L$ and $(b_{k})$ converges to $L$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Limits and Cauchy sequences of reals).

[L1]

Convergence, quantified over rational $ε > 0$ (Limits and Cauchy sequences of reals).

[L2]

Absolute value: for $c > 0$ , $∣ u ∣ < c$ if and only if $- c < u < c$ (Basic properties of the absolute value).

[L3]

Order arithmetic in $R$ : adding a constant preserves $<$ and $\leq$ ; $u < v \leq w$ and $u \leq v < w$ both give $u < w$ ; and $\leq$ is transitive (Order is preserved by adding a constant and by adding inequalities, Complete ordered field (least-upper-bound property), Ordered field).

[L4]

The order on $N$ is total and transitive, so three indices $K_{0}, K_{1}, K_{2}$ admit an index $K$ with $K \geq K_{0}$ , $K \geq K_{1}$ and $K \geq K_{2}$ , and then $k \geq K$ gives $k \geq K_{0}$ , $k \geq K_{1}$ , $k \geq K_{2}$ ( $\leq$ is a linear order on $N$ ).

Proof

technique · direct

1.1

Let $ε > 0$ be rational. By convergence of $(a_{k})$ there is $K_{1}$ with $∣ a_{k} - L ∣ < ε$ for all $k \geq K_{1}$ , and by convergence of $(b_{k})$ there is $K_{2}$ with $∣ b_{k} - L ∣ < ε$ for all $k \geq K_{2}$ .

2.1

By [L2], $∣ a_{k} - L ∣ < ε$ gives $- ε < a_{k} - L$ , that is $L - ε < a_{k}$ , for all $k \geq K_{1}$ ; and $∣ b_{k} - L ∣ < ε$ gives $b_{k} - L < ε$ , that is $b_{k} < L + ε$ , for all $k \geq K_{2}$ .

step 1.1L2L3

3.1

Choose $K$ with $K \geq K_{0}$ , $K \geq K_{1}$ and $K \geq K_{2}$ . For every $k \geq K$ the hypothesis gives $a_{k} \leq x_{k} \leq b_{k}$ , so $L - ε < a_{k} \leq x_{k} \leq b_{k} < L + ε$ , hence $L - ε < x_{k} < L + ε$ , hence $- ε < x_{k} - L < ε$ , hence $∣ x_{k} - L ∣ < ε$ .

step 2.1givenL2L3L4choose

4.1

The rational $ε > 0$ was arbitrary, so for every rational $ε > 0$ there is an index $K$ beyond which $∣ x_{k} - L ∣ < ε$ ; that is, $(x_{k})$ converges to $L$ .

step 3.1L1∎

Remarks

The proof is direct from the definition and does not route through Limits preserve non-strict inequalities. It could not: that lemma assumes the middle sequence converges, which is what is being proved here.
Both bounding sequences must have the same limit. With $a_{k} = - 1$ , $b_{k} = 1$ and any $(x_{k})$ taking values in $[- 1, 1]$ the hypothesis $a_{k} \leq x_{k} \leq b_{k}$ holds and nothing whatever follows, since a bounded sequence need not converge (FALSE: every bounded sequence converges).
The most common use is with $a_{k} = - c_{k}$ and $b_{k} = c_{k}$ for a null sequence $(c_{k})$ : if $∣ x_{k} ∣ \leq c_{k}$ eventually and $c_{k} \to 0$ , then $x_{k} \to 0$ . That special case also follows from A null sequence times a bounded sequence is null whenever $x_{k}$ is presented as a product of a null sequence and a bounded one, but the squeeze needs no such presentation.

Depends on

Used by

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. K. Hunter, An Introduction to Real Analysis, Ch. 3 (standard reference, not scraped)
Squeeze theorem (Wikipedia) (standard reference, not scraped)
T. Tao, Analysis I, 3rd ed., §6.4 (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (standard reference, not scraped)