Let $W$ be a standard, one-dimensional Brownian motion. Let $T\in(0,+\infty)$. Then $$\lim_{\beta\to+\infty}\sup_{0\le t\le T}\left|e^{-\beta t}\int_0^te^{\beta s}\mathrm{d}W_s\right|=0\quad\text{a.s.}$$ Could someone give some hints or proofs?
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