Let $X_n$ be a positive continuous random variable that converges in probability to $0$. I would to prove that
$$E[X_n^{-0.2}] < \infty$$
My approach would be to use a Taylor expansion around a value $a \neq 0$ and then obtain
$$E[a^{-0.2} -0.2 a^{-1.2}(X_n - a) + 1.2*0.2 a^{-2.2}(X_n-a)^2] + o(1) < \infty$$
Since I know that $E[X_n^3] \rightarrow 0$. But I have the feeling that something is wrong.