Prove that the sequence $(p^n)^{∞}_{n=1}$ for p a prime is a null sequence with respect to $| · |_p$.

Question

Prove that the sequence $(p^n)^{∞}_{n=1}$ for p a prime is a null sequence with respect to $| · |_p$.

Here is what I have:

1. a null sequence is a sequence that maps to zero.
2. $| · |_p=p^{-vp(\cdot)}$

So, using (1) and (2) it seems obvious that for $|p^{n}|_p=p^{-vp(p^{n})}$ as $n\rightarrow\infty,p^{-vp(p^{n})}\rightarrow 0 $, which is a null sequence. However, I am having trouble putting this into a coherent proof.

Please help me with suggestion on how to get this started. Thank you.