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?