I've read a proof for $E(X_n^p)\to E(X^p)$ for some $p\ge1$ (where $X,X_1,X_2,...$ are random variables on some probability space $(\Omega,\mathcal{F},P$)).
It uses Minkowski's inequality,
$(E|X^p|)^{1/p}\le(E(|X_n-X|^p))^{1/p}+(E|X_n^p|)^{1/p}$,
and states that we let $n\to\infty$, we get $\liminf\limits_{n\rightarrow\infty} E|X_n^p|\ge E|X^p|$.
And then similarly,
$(E|X_n^p|)^{1/p}\le(E(|X_n-X|^p))^{1/p}+(E|X^p|)^{1/p}$,
and states that we let $n\to\infty$, we get $\limsup\limits_{n\rightarrow\infty} E|X_n^p|\le E|X^p|$.
Is there someone kind enough to give an explicit explanation on why one can conclude those limits (superior and inferior) from the inequality?
Let $\epsilon >0$. $(E|X_n|^{p})^{1/p} > (E|X|^{p})^{1/p}-\epsilon$ for $n$ sufficiently large. Hence $\lim \inf (E|X_n|^{p})^{1/p} \geq (E|X|^{p})^{1/p}-\epsilon$. Let $\epsilon \to 0$. You get $\lim \inf (E|X_n|^{p})^{1/p} \geq (E|X|^{p})^{1/p}$. Now $\lim \inf a_n^{1/p} =(\lim inf a_n)^{1/p}$ for any sequence of non-negative numbers $(a_n)$. This proves the first inequality. The second one is very similar.