# Weak law of large numbers: counterexample for independent but not i.i.d. variables

Can someone please give me an example for sequence $\{X_n\}$ of independent random variables, such that $$E[|X_n|]<5$$ for each n, and such that the weak law of large numbers doesn't hold for it ?

Nothing probabilistic here... Try $X_n=1$ with full probability if $4^k\leqslant n\lt2\cdot4^k$ for some integer $k$, and $X_n=0$ with full probability otherwise.