Let $\{d_n\}$ be a sequence of 0's and 1's and define a sequence of numbers $\{a_n\}$ by $$a_n = d_1\cdot2^{-1}+d_2\cdot2^{-2}+\dots+d_n\cdot2^{-n}.$$ Prove that this sequence converges to a number between 0 and 1.
I've shown that the largest value of $a_n$ occurs when $d_k = 1$ for all $k\le n$ (and that the smallest value occurs when $d_k = 0$ for all $k\le n$). I'm not sure how to show that $\lim a_n \le 1$ when $d_n$ = 1 for all $n$. I know $\{a_n\}$ is non-decreasing. I've tried to use induction, but I quickly hit a wall. I won't type out exactly what I've tried with induction unless needed, just because I have a hunch (possibly erroneously) that induction isn't the way to go.
You are basically done already. Your sequence is bounded by $$ 0 \leq a_n \leq \sum_1^{\infty}2^{-n}=1 $$ You have a monotone, increasing sequence, which is bounded above.Therefore a limit exists. Now you can simply assume the existence of the limit.
For this limit, there holds $$ 0\leq\lim_{n \to \infty} a_n \leq 1 $$ as for all $n \in \mathbb{N}$ there holds $0 \leq a_n \leq 1$. Taking limits preserves "$\leq$".