LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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For fixed $k$ , $\binom{n}{k}/n^k$ tends to $1/k!$

Statement

For each fixed $k\in\mathbb N$ , $\frac{\iota\binom nk}{\iota(n)^k}\longrightarrow\frac1{\iota(k!)} \qquad(n\to\infty),$ where the expression is read for $n\ge\max\{1,k\}$ . For every such $n$ , one also has the uniform bound $0\le \frac{\iota\binom nk}{\iota(n)^k}\le\frac1{\iota(k!)}.$

Facts & Assumptions

Given: A fixed $k\in\mathbb N$ .

[L1]

For $n\ge k$ , $\binom nk\,k!=n^{\underline{k}}$ ( $\binom{n}{k}\,k!\,(n-k)! = n!$ for $k \le n$ ; hence $\binom{n}{k}\,k! = n^{\underline{k}}$ , the quotient $n!/(k!(n-k)!)$ is a natural number, and $\binom{n}{k} = \binom{n}{n-k}$ , The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $\mathbb{N}$ ), and the canonical embedding preserves the products involved (The canonical natural $\iota(n) = n \cdot 1_F$ of a field).

[L2]

Finite products and sums obey Finite sums and finite products, by recursion and Laws of finite sums and finite products, limits obey Algebra of limits: sums, scalar multiples, products and quotients, and $1/\iota(n)\to0$ (For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ ). Canonical naturals are positive and strictly increasing (Canonical naturals are positive and strictly increasing), while multiplication by a positive real preserves order (Sign rules for products and monotonicity of multiplication, Ordered field).

Proof

technique · direct

1.1

For $n\ge\max\{1,k\}$ , $\iota\binom nk/\iota(n)^k=\iota(k!)^{-1}\prod_{j<k}(1-\iota(j)/\iota(n))$ .

L1givenalgebra

2.1

For $j<k\le n$ , strict increase and positivity give $0\le\iota(j)/\iota(n)<1$ , so every factor in step 1.1 lies in $(0,1]$ . Thus the finite product lies in $[0,1]$ , proving the displayed uniform bound.

step 1.1L2

3.1

For each of the finitely many $j<k$ , $\iota(j)/\iota(n)\to0$ ; finite-product limit algebra makes the product in step 1.1 tend to $1$ . Multiplication by the fixed factor $1/\iota(k!)$ yields the limit.

step 1.1L2∎

Depends on

Used by

For every real x, (1+x/n)ⁿ→exp x Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 99 results over 26 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
J. Lebl, Basic Analysis, Analytic Functions (standard reference, not scraped)
J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
J. K. Hunter, An Introduction to Real Analysis, Chapter 10 (standard reference, not scraped)
R. Sedgewick and P. Flajolet, Analytic Combinatorics, Asymptotic Approximations (standard reference, not scraped)

For fixed kkk, (nk)/nk\binom{n}{k}/n^k(kn​)/nk tends to 1/k!1/k!1/k!