So I have a problem and have simplified the part I am confused about below.
If $\sum_{m=1}^{\infty }c < \infty$ and $0 \leq c \leq 1$, then $lim_{n\rightarrow \infty} \sum_{m=n}^{\infty }c= 0$ which implies $c=0$.
My general intuition says that because the sum of infinitely many non-negative c's is less than infinity, than $c=0$ because the sum of an infinitely many positive numbers will always be infinity.
The limit is where I am confused. I feel like the limit will always be $0$ even if $c>0$. It also feels like the limit is not necessary to show $c=0$.
Answer
If $c>0$ then $\sum_{i=1}^{\infty }c= \lim_{n\to \infty }
\sum_{i=1}^{n}c=\lim_{n\to \infty }nc =\infty $
If c=$0$ then $\sum_{i=1}^{\infty }c=0 $
If $c<0 $then $\sum_{i=1}^{\infty }c =-\infty $
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