Let $ X_n $ be sequence of rv´s on $ ( \Omega, F, P ) $. We know that $ E[X_n^2] < \infty, \lim_{n\rightarrow \infty} E[X_n] = \mu \in \mathbb{R} $ and $ \lim_{n \rightarrow \infty} \operatorname{Var}(X_n) = 0 $
Show that $ X_n \overset{P} \rightarrow \mu $
My idea: i have to show that $ \lim_{n \rightarrow \infty} P( \vert X_n - X \vert \geq \epsilon ) = 0 $
in my case i can write : $ \lim_{n \rightarrow \infty} P(\vert X_n - \mu \vert \geq \epsilon) \overset{\mathrm{Tschebychev}} \leq \lim_{n \rightarrow \infty} \frac{\operatorname{Var}(X_n)}{\epsilon^2} =0 $ because of $ \lim_{n \rightarrow \infty} \operatorname{Var}(X_n) = 0 $
So i get the convergence in probability and that $ X_n \overset{P} \rightarrow \mu $