Proving Cauchy Sequences with Totient Theorem

[Dragonfall]

Homework Statement

If p does not divide a, show that $$a_n=a^{p^{n}}$$ is Cauchy in $$\mathbb{Q}_p$$.

The Attempt at a Solution

We can factor $$a^{p^{n+k}}-a^{p^n}=a^{p^n}(a^{p^{n+k}-1}-1)$$. p doesn't divide $$a^{p^n}$$ so somehow I must show that $$a^{p^{n+k}-1}-1$$ is divisible by larger and larger powers of p. I feel it has something to do with the totient theorem, but I can't get it to work.

 

[Dragonfall]

Anyone help?