$P[X_n \in [\alpha, \beta]]\geq 1-\epsilon$ for all $n \in \mathbb{N}$.

Question

Let $(X_n)$ r.v. such that converge in distribution to $X$.

(a) Prove that for each $\epsilon > 0$ exists $\alpha,\beta \in \mathbb{R}, \alpha <\beta$, such that
$$P[X_n \in [\alpha, \beta]]\geq 1-\epsilon \ \ \ \text{for all $n \in \mathbb{N}$}.$$

(b) Let $S_n=\sum_{k=1}^n X_k$. Prove that if $S_n/\sqrt{n}$ converges in distribution to a r.v. $Y$, then $$\frac{S_n}{n} \xrightarrow{P}0.$$

I think (a) is from definition since $X_n$ converge in distribution to $X$ then $\{X_n\}$ is tight and hence for all $\epsilon>0$ there exist $\alpha,\beta \in \mathbb{R}, \alpha <\beta$ such that

$$P[X_n \in [\alpha, \beta]]\geq 1-\epsilon \ \ \ \text{for all $n \in \mathbb{N}$}.$$

Is this right?

For the second statement I'm stuck. Any help? Thanks in advance.