Show that if $x\in Q_p$ and $x\neq 0$ then there exixts $x^{-1}\in Q_p$
I tried to constuct a y such that xy=1. By the definition of $Q_p$, $x=a_{-l}p^{-l}+a_{-l+1}p^{-l+1}+...$ and I want to find $ (b_{-l},b_{-l+1},...)$ such that $xy=(a_{-l}p^{-l}+a_{-l+1}p^{-l+1}+...)(b_{-l}p^{-l}+b_{-l+1}p^{-l+1}+...)=1$ But It turns out that this cannot be solved.