Support of closed random walk on $\mathbb Z$

One way to do this is as follows. We have to show that $$P(M_n\ge x|S_n=0)\to1$$ (as $n\to\infty$) if $x=o(\sqrt n)$, where $S_n$ is the position of the walk at time $n$ and $M_n:=\max_{0\le k\le n}S_k$. By the reflection principle (see e.g. Theorem 0.8) and the de Moivre--Laplace theorem , for natural $x$ such that $x=o(\sqrt n)$, $$P(M_n\ge x,S_n=0)=P(S_n=2x)\sim P(S_n=0),$$ whence $$P(M_n\ge x|S_n=0)=\frac{P(M_n\ge x,S_n=0)}{P(S_n=0)}\to1,$$ as desired.