Probability Bridge

For fixed \(\alpha\) and \(\beta\) consider the stochastic differential equation
\[
dY(t)=\frac{\beta-Y(t)}{1-t} dt + dB(t) ~,~~ 0\leq t < 1 ~,~~Y(0)=\alpha.
\]
Verify that \(\lim_{t\to 1}Y(t)=\beta\) with probability one. ( This is called the Brownian bridge from \(\alpha\) to \(\beta\).)
Hint: In the problem “Solving a class of SDEs“,  you found that this equation had the solution
\begin{equation*}
Y_t = a(1-t) + bt + (1-t)\int_0^t \frac{dB_s}{1-s} \quad 0 \leq t <1\; .
\end{equation*}
To answer the question show that
\begin{equation*}
\lim_{t \rightarrow 1^-} (1-t) \int_0^t\frac{dB_s}{1-s} =0 \quad \text{a.s.}
\end{equation*}