Random Walk Upper-Bound. For any symmetric Random Walk, absolute value of partial sum $S_n$ never exceeds $\sqrt{2 \pi n}+\sqrt{\frac{\pi}2}$? Is it possible to prove that for any symmetric Random Walk, the absolute value of its partial sum $S_n$ never exceeds $\sqrt{2 \pi n}+\sqrt{\frac{\pi}2}\quad\large?$

I run some simulation, clearly for initial value it is not true, but for large $n$ it seems true that it is a real absolute bound.

My motivation was this.
The average number of zeros for Sn is Sqrt[2/Pi] for every interval
2n+1. So if after k^2 steps the value of Sn is greater than (2k+1)*Sqrt[Pi/2]
than in the next interval 2k+1 the possibility for Sn to become zero would be 
impossible. Something wrong?
