Alphabeta MathHow 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 logarithm grows more slowly than every positive real power
Statement
For every ,
Facts & Assumptions
Given: A real .
tends to as , and its range is (The exponential tends to at and to at , The natural logarithm as the inverse of the exponential function).
for , and algebra of limits permits substitution through the displayed identities (Real powers for positive bases, with the zero-base positive-exponent convention, Algebra of limits: sums, scalar multiples, products and quotients).
Proof
Put . As , the inverse relation and [L1] give .
By [L3], .
The right-hand side tends to by [L2], which proves the claim.
Depends on
- The exponential dominates every fixed nonnegative integer power at $+\infty$
- Real powers for positive bases, with the zero-base positive-exponent convention
- The natural logarithm as the inverse of the exponential function
- The exponential tends to $+\infty$ at $+\infty$ and to $0$ at $-\infty$
- Algebra of limits: sums, scalar multiples, products and quotients
Used by
- The hereditary class of all finite graphs does not have the Erdős–Hajnal property Corollary
- The universal logarithmic Ramsey guarantee cannot be replaced by any universal positive power Counterexample
- Concrete logarithmic, polynomial, and exponential growth comparisons Example
- xˣ tends to one as x tends to zero from the right Example
- For all positive k,ℓ, some finite graph has girth greater than ℓ and chromatic number greater than k Theorem
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
- J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)