Let $\{c_j : j \ge 1\}$ be a sequence of positive numbers and $s_m = \sum_{j=1}^m c_j$ be a sequence of partial sums. Suppose $s_m \le k $ for all $m \ge 1$. What is $\lim_{m \to \infty} s_m$? Show that this is a Cauchy sequence.
I think I know how to show that it is Cauchy pretty easily. From the way it is defined, the sequence has to be strictly increasing and it is bounded, which quickly gives the Cauchy property.
However, I'm not sure I see how to show what the limit actually is. My immediate guess would be that it is $k$, but that's just a hunch.... I feel like I might need to do something with subsequences, but that's also just a feeling.