Let $X_i$ be a sequence of random variables such that $E[X_i]=0$, then using Checbychev's Inequality:
$$Pr(|\frac{1}{n}\sum{X_i}|>\epsilon)\leq\dfrac{E[\frac{1}{n}\sum{X_i}]}{\epsilon}=\dfrac{\frac{1}{n}\sum{E[X_i]}}{\epsilon}=0$$
Now since the LHS is a probability, it must hold that:
$$Pr(|\frac{1}{n}\sum{X_i}|>\epsilon)=0$$
This is obviously false and would only make sense if a limit was applied to the LHS due to the weak law of large numbers. What is wrong with my working?