How to derive $\sup_{|s-t|<a}|N(0,\frac{1}{2}(e^{2t}-e^{2s}))|=O_p((a\log a)^{1/2})$

Question

I want to derive the sup norm of $\int_s^te^udW(u)$, where $t-s<a, W(u)$ is standard Brownian motion. As $e^u$ is not a random variable, $\int_s^te^udW(u)$ is a normal variable with mean 0 and variance $\int_s^te^{2u}du=\frac{1}{2}(e^{2t}-e^{2s})$, then what to do next to obtain the the following order? $$ \sup_{|s-t|<a}|\int_s^te^udW(u)|=\sup_{|s-t|<a}|N(0,\frac{1}{2}(e^{2t}-e^{2s}))|=O_p((a\log a)^{1/2}). $$