Consider a symmetric random walk. Let $M_n = \max\{0,S_1,\cdots,S_n\}$. I want to find
$$\lim_{n\to\infty} Pr( M_n / \sqrt{n} \leq x) $$
I know how to calculate $Pr(M_n = m)$ and how it is connected to the $Pr(S_n = k)$ and I want to use it in my proof, but do not really now how. My best bet is using Central Limit theorem. Would appreciate if someone can help me with tying these two things together.
By the reflection principle, for each $n, k \geq 0$ we get
$$ \operatorname{Pr}(M_n \geq k) = \operatorname{Pr}(S_n \geq k) + \operatorname{Pr}(S_n > k). $$
From this, it is not hard to deduce that, for $x > 0$,
$$ \lim_{n\to\infty} \operatorname{Pr}(M_n / \sqrt{n} \leq x) = 1 - 2 \lim_{n\to\infty} \operatorname{Pr}(S_n / \sqrt{n} \geq x) = 2\operatorname{Pr}(Z \leq x) - 1, $$
where $Z \sim \mathcal{N}(0, 1)$ is a standard normal variable.