Limit of a Wiener integral

The set of functions of class $\mathcal C^1$ on $[0,T]$ is dense into the set of continous functions on $[0,T]$. Since $t\mapsto B_t$ is continuous we have, a.s. for every $\epsilon >0$, there exists $\left ( B_t ^\epsilon\right)_{t \in \left[0,T \right] }$ of class $\mathcal C^1$, such that

$$\sup _ {t \in \left[0,T \right] } \left | B_t-B_t^\epsilon\right | < \epsilon.$$

Since we have by Itô's lemma that

$$ \int_0 ^t e^{\alpha s} ~dB_s = e^{\alpha t}B_t - \int_0 ^t B_s \alpha e^{\alpha s}ds$$

and also by integration by parts

$$ \int_0 ^t e^{\alpha s} ~dB_s ^\epsilon= e^{\alpha t}B_t ^\epsilon - \int_0 ^t B_s ^\epsilon \alpha e^{\alpha s}ds\qquad\text{a.s.}$$

this shows that

\begin{align} \left |\int_0 ^t e^{\alpha s} ~dB_s - \int_0 ^t e^{\alpha s} ~dB_s ^\epsilon\right| &=\left | e^{\alpha t}B_t -e^{\alpha t}B_t ^\epsilon - \int_0 ^t B_s \alpha e^{\alpha s}ds + \int_0 ^t B_s ^\epsilon \alpha e^{\alpha s}ds\right| \\ &\leq e^{\alpha t}\left |B_t -B_t ^\epsilon\right| + \int_0 ^t \alpha e^{\alpha s}\left|B_t -B_t ^\epsilon\right|ds \\ &\leq \epsilon e^{\alpha t} + \epsilon \int_0 ^t \alpha e^{\alpha s} ds \leq 2\epsilon e^{\alpha t}\qquad\text{a.s.}\end{align}

so $$\left |e^{-\alpha t}\int_0 ^t e^{\alpha s} ~dB_s - e^{-\alpha t}\int_0 ^t e^{\alpha s} ~dB_s ^\epsilon\right| \leq 2\epsilon\qquad\text{a.s.} $$

Furthermore,

$$ \left| e^{-\alpha t}\int_0 ^t e^{\alpha s} ~dB_s ^\epsilon\right|\le\frac {\left \| \dot{B^\epsilon}\right\|_\infty }{\alpha} \left( 1 - e^{-\alpha t} \right)\le\frac {\left \| \dot{B^\epsilon}\right\|_\infty }{\alpha}\qquad\text{a.s.} $$ Summing these and considering the supremum over $t\in[0,T]$, one gets $$ \sup _ {t \in \left[0,T \right] }\left |e^{-\alpha t}\int_0 ^t e^{\alpha s} ~dB_s \right| \leq 2\epsilon+\frac {\left \| \dot{B^\epsilon}\right\|_\infty }{\alpha}\qquad\text{a.s.} $$ hence $$ \limsup_{\alpha\to\infty} \sup _ {t \in \left[0,T \right] }\left |e^{-\alpha t}\int_0 ^t e^{\alpha s} ~dB_s \right| \leq 2\epsilon\qquad\text{a.s.} $$ This holds for every $\epsilon\gt0$ hence $$ \lim_{\alpha\to\infty} \sup _ {t \in \left[0,T \right] }\left |e^{-\alpha t}\int_0 ^t e^{\alpha s} ~dB_s \right| =0\qquad\text{a.s.} $$