Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Use the Borel-Cantelli lemma to show that the running maximum \(M_{n}\) := \(\sup _{0 \leqslant t \leqslant n} B_{t}\) cannot grow faster than \(C \sqrt{n \log n}\) for any \(C>2\). Use this to show that

\[\varlimsup_{t ightarrow \infty} \frac{M_{t}}{C \sqrt{t \log t}} \leqslant 1 \quad \text { a.s. }\]

Show that \(M_{n}(\omega) \leqslant C \sqrt{n \log n}\) for sufficiently large \(n \geqslant n_{0}(\omega)\).