The weak law needs only uncorrelatedness

Prove that if $X_1,X_2,\dots$ are uncorrelated with common mean $\mu$ and variance at most $\sigma^2$, then $\bar X_n\xrightarrow{\P}\mu$, with the explicit bound $\P(|\bar X_n-\mu|\ge t) \le \sigma^2/(nt^2)$. Then give an example of uncorrelated but dependent variables to show the hypothesis is genuinely weaker than independence.