Convergence in Lp

Let $\{X_n\}$ be a sequence of random variables and $S_n:=\sum_{j=1}^n X_j$. Proof that if $X_n\overset{\mathcal{L}_p}{\to}0$ for some $p\geq 1$, then $n^{-1}S_n\overset{\mathcal{L}_p}{\to} 0$ but the converse is not true in general.

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