Given the function $Y(t) = a(1-t) + bt + (1-t)\int_{[0,t]}\frac{dW_s}{1-s}$ with $t \in [0,1)$ we are asked to show the limit as $t \to 1$ of $Y(t) = b$. $W_t$ is a brownian motion with distribution N(0,t).
The first 2 terms of $Y(t)$ are trivial but the last stochastic piece I am having some trouble proving. The result would be sufficient if I can show the limit $t \to 1$ of term $(1-t)\int_{[0,t]}\frac{dW_s}{1-s}$ goes to 0 a.s., but I am not clear how to show this.