This is a homework problem. If $X_n$ converges in probability to 1, show $X_n^{-1}$ converges in probability to 1.
My attempt:
$$\begin{align*} P(|X_n^{-1}-1| > \epsilon) &= P(|X_n^{-1}-X_n + X_n-1| > \epsilon)\\ &\leq P(|X_n^{-1} - X_n| > \epsilon/2) + P(|X_n - 1| > \epsilon/2)\\ &= \end{align*} $$
I know I can bound the 2nd term, but I am not sure how to bound the first term. Perhaps another approach is necessary. Any suggestions would be appreciated.