LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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A minimizing sequence in a convex set is Cauchy

Statement

Let $C$ be a nonempty convex subset of a real or complex inner-product space, let $x$ be a vector and put $d = \inf_{c \in C} ∥ x - c ∥$ . If $c_{n} \in C$ is a sequence with $∥ x - c_{n} ∥ \to d$ , then $(c_{n})$ is a Cauchy sequence.

Facts & Assumptions

[A1]

A subset $C$ is convex when $(1 - t) u + t v \in C$ for all $u, v \in C$ and $0 \leq t \leq 1$ (Convex sets and continuous real-hyperplane separation in a normed space).

[A2]

If $d = \inf S$ then $d \leq s$ for every $s \in S$ (Greatest lower bound (infimum)).

[A3]

Every inner-product norm satisfies the parallelogram law (The parallelogram law).

[A4]

The norm is induced by the pairing, in particular $∥ u - v ∥$ is the distance between $u$ and $v$ and $∥ u + v ∥ = ∥ v + u ∥$ (Real and complex inner-product spaces and their induced length).

Proof

technique · direct

Given: A nonempty convex set $C$ , a vector $x$ , the number $d = \inf_{c \in C} ∥ x - c ∥$ and a sequence $c_{n} \in C$ with $∥ x - c_{n} ∥ \to d$ .

1.1

Put $u_{n} = x - c_{n}$ ; for all $m, n$ the midpoint $(c_{m} + c_{n}) / 2$ lies in $C$ by convexity, so $∥ \frac{1}{2} (u_{m} + u_{n}) ∥ = ∥ x - \frac{1}{2} (c_{m} + c_{n}) ∥ \geq d$ because $d$ is a lower bound of the distances from $x$ to points of $C$ .

A1A2A4

2.1

The parallelogram law applied to $u_{m}, u_{n}$ gives $∥ u_{m} - u_{n} ∥^{2} = 2 ∥ u_{m} ∥^{2} + 2 ∥ u_{n} ∥^{2} - 4 ∥ \frac{1}{2} (u_{m} + u_{n}) ∥^{2} \leq 2 ∥ u_{m} ∥^{2} + 2 ∥ u_{n} ∥^{2} - 4 d^{2}$ .

step 1.1A3algebra

3.1

Given $ε > 0$ , convergence $∥ u_{n} ∥ \to d$ provides $N$ with $2 ∥ u_{n} ∥^{2} - 2 d^{2} < ε^{2} / 2$ for all $n \geq N$ , and then step 2.1 gives $∥ c_{m} - c_{n} ∥^{2} = ∥ u_{m} - u_{n} ∥^{2} < ε^{2}$ for all $m, n \geq N$ ; hence $(c_{n})$ is Cauchy.

step 2.1algebra∎

Depends on

Used by

Projection onto a nonempty closed convex set Theorem

Dependency tree · two levels

12 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

Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 1.44, pp.40–41 (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 178 (standard reference, not scraped)