= Solution
For the <simple symmetric random walk>, the increments have mean $0$ and <variance> $1$. Use the polygonal interpolation from (b). The maximum functional is Lipschitz in the <supremum norm>, since
$$
\left|\max_{0\leq s\leq1}g(s)-\max_{0\leq s\leq1}h(s)\right|\leq\|g-h\|_\infty.
$$
Each interpolating segment is linear, so its maximum occurs at a grid endpoint. The <continuous mapping theorem> applied to the <Donsker invariance principle> gives
$$
\frac1{\sqrt n}\max_{0\leq j\leq n}S_j\ \Rightarrow\ \max_{0\leq s\leq1}B_s.
$$
The limiting maximum has no atom at $a>0$ by the <Brownian reflection principle>, so the closed-tail <probabilities> converge at this threshold. Including $j=0$ does not change the event because $a>0$. Therefore the <random-walk maximum limit from Donsker invariance> is
$$
\boxed{\lim_{n\to\infty}\mathbb P\left(\max_{j\leq n}S_j\geq a\sqrt n\right)=2\bigl(1-\Phi(a)\bigr),\qquad a>0.}
$$
Here $\Phi$ is the <standard normal distribution function>. The normalization and the positive-threshold condition are those in the PDF; the TeX aid's $\sqrt t$ is not the printed expression.
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