- #1

- 1,456

- 44

## Homework Statement

Prove that if ##a > 1##, then the sequence ##a^n## is unbounded.

## Homework Equations

## The Attempt at a Solution

We need to show that ##\forall M \in \mathbb{R}~\exists N \in \mathbb{N},~~ a^n > M##. To do this I thought maybe we could let ##N = \operatorname{ceil}(\log_a(M))+1##, so that we are guaranteed that ##a^N > M##, but I don't think this will work for several reasons. One, ##M## could be negative. And two, I formally haven't defined the notion of logarithm yet. What approach should I take then?