Let $X_n, n \in \mathbb{N}$ be a sequence of random variables with $\lim_{n \rightarrow \infty}\mathbb{E}[X_n^2] = 0$. Show that $X_N \rightarrow_d X$ with $X = 0$ almost surely, as $n \rightarrow \infty$.
I was thinking about showing $P$-convergence. This would already imply convergence in distribution. I already noticed $-\mathbb{E}[X^2] \leq \mathbb{E}[X] \leq \mathbb{E}[X^2]$. Now I need to show for every $\epsilon > 0$:
$$P[|X_n - 0| > \epsilon] \rightarrow 0 \text{ as } n \rightarrow \infty$$
and sure I can somehow show this using the fact above. Thank you.
Note that $P(|X_n|>\epsilon)\leq P(X_n^2>\epsilon^2)\leq \frac{1}{\epsilon^2}E[X_n^2\times 1_{\{X_n^2>\epsilon^2\}}]\leq \frac{1}{\epsilon^2}E[X_n^2]\to 0$.