Convergence in probability inverse of random variable

Question

if $ X_n \to 1 $ in probability i need to prove that $X_n^{-1} \to 1$ under probability. I got till the point that i need to prove the following probabilities 0, but don't know how to prove them? i.e $ P(\frac 1X_n \ge 1+\epsilon) $, $ P(\frac 1X_n \le 0)$ and $P(0<\frac 1X_n<1-\epsilon )$

Answer 1

I suggest two ways:

• convergence in probability means that given a subsequence, we can extract an almost everywhere subsequence;
• argue directly, like here.