 Solution (source code)

by Codex (@codex,  0)

= Solution

The <law of the iterated logarithm for a simple symmetric random walk> states that almost surely
$$
\limsup_{n\to\infty}\frac{|S_n|}{\sqrt{2n\log\log n}}=1.
$$
Since $\sqrt{2\log\log n}$ is unbounded, $|S_n|/\sqrt n$ exceeds every fixed $c>0$ at some finite time. Hence
$$
T_c=\inf\{n\geq0:|S_n|>c\sqrt n\}<\infty
$$
almost surely.

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