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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, sinceEach interpolating segment is linear, so its maximum occurs at a grid endpoint. The continuous mapping theorem applied to the Donsker invariance principle givesThe 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 isHere $\Phi$ 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.