Let $Y_1, Y_2, Y_3,\ldots,Y_n,\ldots$ be iid and bounded random variables with $E[Y_1]=0$. Define $X_n = Y_1+Y_2+ \cdots + Y_n$.
If $\Pr(Y_1 \neq 0) \gt 0$, then $ \limsup X_n = \infty$ with probability $1$.
The conclusion seems intuitive, but how would I approach it rigorously?