CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The elementary numerical bound $2 < e < 3$

Statement

$2 < e < 3.$

Facts & Assumptions

Given: $e = \sum_{k \geq 0} 1 / ι (k!)$ (The real exponential function and the number $e$ by a power series).

[L1]

The factorial tail bound is A geometric bound for tails of the exponential series.

Proof

technique · direct

1.1

The first three terms give $e > 1 + 1 + 1 / 2 > 2$ .

given

1.2

The term at index $2$ is $1 / 2$ . Apply [L1] with $x = 1, N = 2$ : the tail from index $3$ onward is at most $2 / 3! = 1 / 3$ . Thus the whole tail after index $1$ is at most $1 / 2 + 1 / 3 = 5 / 6 < 1$ , and hence $e < 2 + 1 = 3$ .

L1algebra

2.1

Combining the strict bounds gives the claim.

step 1.1step 1.2∎

Depends on

Used by

The convergence (1+x/n)ⁿ→exp x is not uniform on ℝ Counterexample
Integral bounds alone give 2<e<4; the sharper published bound is 2<e<3 Example

Dependency tree · two levels

23 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

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)
LSU MATH 7230, Homework 1 (standard reference, not scraped)
University of Michigan MATH 295 notes (standard reference, not scraped)

The elementary numerical bound 2<e<32<e<3