Random-walk maximum limit from Donsker invariance ($n^{-1/2}{\max }_{j\le n}{S}_j\Rightarrow {\sup }_{s\le 1}{B}_s$)

For a random walk with centered independent identically distributed increments of variance one, the maximum of its polygonal interpolation is the maximum at its grid vertices. The maximum functional is Lipschitz for the supremum norm, so the continuous mapping theorem applied to the Donsker invariance principle proves the displayed convergence in distribution. The Brownian reflection principle then gives upper-tail limits $2(1-\Phi (a))$ at all $a>0$, since the limiting maximum has no atom there.