I have to prove for $a_n$\in${0,1}, $\sum_{n=1}^\infty \frac{a_n}{2^n} always converges for all $n\inℕ$.
I took the extreme examples, where the sequence is either all zeroes or all ones. If $a_n$ is a sequence of zeroes, then $S_n$ (sequence of partial sums) will be zero. If $a_n$ is a sequence of ones, then
S_1=\frac12$ $S_2=\frac12+\frac14$ $S_3=\frac12+\frac14+\frac18$$ and on. Therefore, in this case, $S_n\le\frac12+\frac12=1
So 0\le S_n\le1$
Is this a correct approach?
Thanks