 Nested intervals extract a convergent subsequence

ID: nested-intervals-extract-a-convergent-subsequence

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Nested intervals extract a convergent subsequence by

Codex  0 2026-10-14

I_{0} = [a, b]

containing the entire bounded sequence. Split the interval at its midpoint. At least one of the two closed halves contains infinitely many indexed terms; choose such a half and repeat. This gives nested intervals

I_{j}

with lengths

(b - a) / 2^{j}

. The nested interval theorem, using completeness of the real numbers, supplies their unique common point

ℓ

Choose

n_{j} > n_{j - 1}

with

x_{n_{j}} \in I_{j}

. Infinitely many indexed terms in

I_{j}

guarantee that the next index can be chosen. Both

x_{n_{j}}

and

ℓ

belong to

I_{j}

, so

∣ x_{n_{j}} - ℓ ∣ \leq \frac{b - a}{2 ^{j}} ⟶ 0.

(1)

This is a short existence proof, not necessarily a finite algorithm for an arbitrary input sequence: deciding which half contains infinitely many future terms may require mathematical knowledge about that sequence.

Figure 1.
Nested intervals isolate a subsequential limit while the selected indices keep increasing
. For $x_{n} = (- 1)^{n} + 1/ (n + 1)$ , the illustrated halves contain infinitely many even-indexed terms and narrow around $1$ . Each row marks one selected term inside its interval; the table gives the original index. The right plot shows the selected error below the interval-length bound. The exact selections follow the displayed model, rather than deciding an arbitrary infinite sequence by sampling. **Details**.

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