Basic Analysis I — Real Analysis · Problem 1.5.8

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · archimedean-property

Problem

Prove that if

b \in \mathbb{N}

and

b \geq 2

, then for every

\epsilon > 0

, there is an

n \in \mathbb{N}

such that

b^{-n} < \epsilon

. Hint: One possibility is to first prove that

b^n > n

for all

n \in \mathbb{N}

by induction.