If $X_n$ converges in distribution to $X$ and $E[X^2]$ is finite, could we have $E[X_n^2]$ is finite?

Question

I only know 1) $X_n$ converges in distribution to $X$; 2) $X_n$ is bounded by a constant $M$, and 3) $\mathbb{E}[X^2]$ is finite, could I get $\mathbb{E}[X_n^2]$ is finite?

Moreover, if $\mathbb{E}[X^k]$ is finite, could we have $\mathbb{E}[X_n^k]$ is finite?

I know that we CANNOT get $\mathbb{E}[X_n]\rightarrow \mathbb{E}[X]$. But could we get $\mathbb{E}[X_n]$ is finite if $\mathbb{E}[X]$ is finite?

Thank you.

Answer 1

$$E[X_n^k]\leq E[M^k]=M^k<\infty$$

Answer 2

The revised statement is false. Let $Y$ be any random variable with $EY^{2}=\infty$, $X=0$ and $X_n=\frac Y n$. Then $X_n \to X$ in distribution (in fact almost surely) and $EX^{2}=0<\infty$ but $EX_n^{2}=\infty$ for all $n$.