CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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The volume of the unit $n$ -ball tends to zero with dimension

Statement

$V_{n} (1) \to 0$ as $n \to \infty$ . The limit is through the positive integers.

Facts & Assumptions

Given: The positive sequence of unit-ball volumes $V_{n} : = V_{n} (1)$ .

[F1]

For every $n \geq 1$ , $V_{n} = π^{n / 2} / Γ (n / 2 + 1)$ (The closed form for the volume of the unit $n$ -ball).

[F2]

For every $s > 0$ , $Γ (s + 1) = s Γ (s)$ (The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ ).

[F3]

If $∣ r ∣ < 1$ , then $r^{k} \to 0$ (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ ).

Proof

technique · direct

1.1

From [F1] and [F2], $V_{n + 2} / V_{n} = 2 π / (n + 2)$ for every $n \geq 1$ .

F1F2algebra

2.1

Choose an integer threshold after which $2 π / (n + 2) \leq 1 / 2$ . Along each parity chain, step 1.1 then bounds every later volume by a fixed initial volume times successive powers of $1 / 2$ , which tend to zero by [F3].

step 1.1F3algebra

3.1

Both the odd-dimensional and even-dimensional subsequences tend to zero, so their interleaving $(V_{n})$ also tends to zero.

step 2.1∎

Depends on

Used by

FALSE: unit-ball volume increases with dimension False statement

Dependency tree · two levels

31 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

Sheldon Axler, Measure, Integration & Real Analysis, §5C, Exercise 12(a) (standard reference, not scraped)

The volume of the unit nn-ball tends to zero with dimension