I have found an old exercise that seems very interesting: let $X\_{1},X\_{2},...$ be i.i.d. bernoullian random variables with $\mathbb{P}(X\_{n}=1) = \mathbb{P}(X\_{n}=0) = 1/2$. Define $S\_{n} := X\_{1} + ... + X\_{n}$. It is to show that the random variable

$$
M := \sup\_\limits{n \in \mathbb{N}}\frac{S\_{n}}{n}
$$
with probability $1$ only takes rational values in the intervall $(1/2,1]$. Anyone has an idea, how to prove it?