Let $W_t$ be a Wiener process. I want to show that: $$ \sum_{n=1}^{\infty} P(\sup_{t\in[n, n+1]} |W_t - W_n| \geq n^{\frac{2}{3}}) < \infty.$$
My attempt:
By Doob's inequality $$P\big(\sup_{t\in[n, n+1]} |W_t - W_n| \geq n^{\frac{2}{3}}\big) = P\big(\sup_{t\in[0, 1]} |W_t| \geq n^{\frac{2}{3}}\big) = P\big(\sup_{t\in[0, 1]} \exp{(|W_t|)} \geq \exp{(n^{\frac{2}{3}})}\big) \leq \frac{\mathbb{E}\big[\exp(|W_1|)\big]}{\exp(n^{\frac{2}{3}})}.$$ Hence $$ \sum_{n=1}^{\infty} P(\sup_{t\in[n, n+1]} |W_t - W_n| \geq n^{\frac{2}{3}}) < \infty.$$ Is my reasoning correct?