Upper law of the iterated logarithm (${\mathrm{limsup}}_n\frac{S_n}{\sqrt{2n\log \log n}}\le 1$)

For a simple symmetric random walk, geometric blocking and the Borel-Cantelli first lemma yield the upper estimate ${\mathrm{limsup}}_n{S}_n/\sqrt{2n\log \log n}\le 1$ almost surely. Equality requires a separate lower-bound argument.