Assume a $S\_n=\sum\_{i=1}^n X\_i$ fulfills a weak law of large numbers, i.e. for every $\epsilon>0$,
$$P\left(|\tfrac{1}{n}S\_n - \mu| > \epsilon\right) \rightarrow 0$$
Does there exist a sequence $(\epsilon\_n)\_n$ with $\epsilon\_n \rightarrow 0$ such that
$$P\left(|\tfrac{1}{n}S\_n - \mu| > \epsilon\_n\right) \rightarrow 0$$
holds? Or do I need to make further assumptions about the $X\_i$s?