Proving $\sup_n\mathbb{E} {[|X_n|^2]}\le \left( \sup_n\mathbb{E} {[|X_n|^p]}\right)^{2/p}$
For $a_n\geq0$, $\alpha >0$, $(\sup_na_n)^\alpha=\sup_{n}a_n^\alpha$, so we only need to prove $$(E|X_n|^2)^p\leq (E|X_n|^p)^2$$ for all $n$. By applying Jensen's inequality with the convex function $\phi(.)=|.|^{p/2}$, $$(E|X_n|^2)^p= \Big(\big(E|X_n|^2\big)^{p/2}\Big)^2\leq E\Big((|X_n|^2)^{p/2}\Big)^2=(E|X_n|^p)^2.$$