Problem: Let $X_1,X_2,\dots$ be i.i.d. $\text{Bernoulli}(1/2)$ random variables.
$\textbf{(a)}$ Show that the sequence $Y_n=\displaystyle\sum_{k=1}^n\frac{X_k}{2^k}$ converges with probability one.
$\textbf{(b)}$ Let $Y=\lim\limits_{n\to\infty}Y_n.$ Show that $Y\thicksim\text{Unif}(0,1)$ by computing $P\left(Y\leq\frac{k}{2^n}\right)$ for $0\leq k\leq2^n$, and then using the density of the dyadic rationals.
Attempt: We begin with part (a). Fix $\omega\in\Omega$. Then $X_k(\omega)=0$ or $X_k(\omega)=1$ for any $k\in\mathbb N$. Therefore,
$$Y_n(\omega)=\sum_{k=1}^n\frac{X_k(\omega)}{2^k}\leq\sum_{k=1}^\infty\frac{1}{2^k}<\infty.$$
Since $Y_n(\omega)$ is a uniformly bounded nondecreasing sum of positive real numbers, it must converge to a limit. Since $\omega\in\Omega$ was arbitrary it follows that $Y_n$ converges with probability one.
For part (b), I found an approach using moment generating functions, in the question https://math.stackexchange.com/a/1269084/595519, but I am having problems coming to grips with the approach indicated above for finding the limiting CDF.
Could someone give me a heads-up on how to approach the problem in part (b)? Any thoughts on part (a) are also much welcomed.
Thank you very much for your time.
