# The eventual domination order

### The mind-expanding property of the eventual domination order—every countable family of functions is strictly bounded. You cannot climb a simple ladder to the top.

The order of *eventual domination* *f* ≤* *g* holds of functions *f*, *g *: ℕ → ℕ when *f*(*n*) ≤ *g*(*n*) for all sufficiently large *n*, which is a precise compact way of saying that there is a natural number *N* such that *f*(*n*) ≤ *g*(*n*) for all *n* ≥ *N*. In other words, *f* ≤* *g* holds if eventually *g*(*n*) is at least as large as *f*(*n*). We similarly define the strict version *f* <* *g*, which means that *f*(*n*) < *g*(*n*) for all sufficiently large *n.*

This will be just a quick taste of the eventual domination order for Panorama of Logic. Some readers may be interested to read the longer gentle introduction to this topic that I provide in my essay The Orders of Infinity, written for The Book of Infinity.

I should like to prove Hausdorff’s theorem, identifying a remarkable feature of the eventual domination order. Namely, every countable sequence of functions is strictly bounded in the order of eventual domination. You cannot climb a simple ladder to the top.

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