P-th sum of a sequence of r.v. converging to p-th mean a.s.? Does this convergence hold in probability?

Question

I have no idea what to show in this task. It looks like a form of LLN but for p-th moments and I couldn't find any statements linking this.

Given a sequence of r.v $X_n$, which are uncorrelated and identically distributed and in $L^{2p}$ on a given probability space and p $ \in [1, \infty] $. Does it hold:

$$\frac{1}{n}\sum_{k=1}^{\infty}|X_k|^p \xrightarrow{n\to\infty} E[|X_1|^p]\; \mathrm{a.s.}$$ Does it hold in probability? So does $$\frac{1}{n}\sum_{k=1}^{\infty} |X_k|^p \xrightarrow{P} E[|X_1|^p]$$ hold.

Thanks for yor help.