Let $(B_t)_{t\geq0}$ be a standard Brownian motion in one dimension. Show that $t^\alpha\mathbb{P}(B_s\leq1,\forall s\leq t)$ converges to a finite positive constant as $t\to\infty$ for some $\alpha$ and find the value of $\alpha$ and the constant.
My attempt so far: Using the reflection principle, I believe that we have:
$$\begin{aligned}t^\alpha\mathbb{P}(B_s\leq1,\forall s\leq t)&=t^\alpha\mathbb{P}(\sup_{0\leq s\leq t}B_s\leq1) \\&=t^\alpha\{1-\mathbb{P}(\sup_{0\leq s\leq t}B_s>1)\}\\&=t^\alpha\{1-2\mathbb{P}(B_t>1)\}\\&=t^\alpha\{2\Phi(1/\sqrt{t})-1\}\end{aligned}$$ where $\Phi$ gives the standard normal cumulative distribution function. I can't see how to progress from here, as the right hand term of course tends to 0 and the left hand term tends to infinity for $\alpha>0$ - any advice would be greatly appreciated!