Let $(c_n)$ be a sequence with $c_i ∈ ${$0, 1$} for all $i$. The series from $n=0$ to infinity, $2c_n/3^{n+1}$. This series converges.
Now, for each $(c_n)$, the series is an element of $C$. I need to use this fact to show that $C$ is an uncountable set.
I have a general idea of what I'm expected to show but I'm not sure how to do it:
For each $(c_n)$, series with the terms $2c_n/3^{n+1}$ converge to different elements in the set $C$. And different starting sequences of $(c_n)$ make up different series so they lead to different convergence numbers in the set.
Here's my thoughts on the problem:
I need to prove that different starting sequences lead to different series. Then, I need to show that there are uncountably many such sequences. This will prove $C$ is an uncountable set.
So if I show there are uncountable many different sequences $(c_n)$, then I can show that there are uncountably many values the series can converge to in the set. So, the set is uncountable.
(I know there are a lot of cantor problems on here, but I'm having a hard time finding one that relates to my problem here. Many of the questions and posts are much more advanced compared to what I'm doing here, so I'm not really able to understand what is going on there. I'm not studying the cantor set, this is just a problem that uses it as context for an analysis problem on sequences and series.)