Let $X_n$ and $X$ be random variables for which $EX_n^p \rightarrow EX^p <\infty$ for $p\in \{1, \ldots, P\}$. $\{X_n\}$ is bounded in probability (this can be shown noting that $EX_n^2 = O(1)$ and applying Markov's inequality), hence by Prokhorov's theorem, each subsequence has a further subsequence, say $\{Y_n\}$, that converges weakly to a limit $Y$. We can show that the moment convergence is also true for $Y_n$: $EY_n^p \rightarrow EY^p <\infty$ for $p\in \{1, \ldots, P\}$.
The claim is that "the moments of $Y$, for $p\in \{1, \ldots, P\}$, are identical to the moments of $X$". How can I show the veracity of this claim?