If $\sum\limits_{t=1}^{\infty}R_t$ is finite with $R_t\geq 0$, does $\lim_{t\rightarrow \infty}R_t=0$?

Question

This question arises from my study of economic models with an infinite time horizon in which production is constrained by a finite initial stock of a non-renewable resource.

Given $\sum\limits_{t=1}^{\infty}R_t = S$ where $S$ is finite and $R_t\geq 0\,\forall t$, can it be inferred that $\lim_{t\rightarrow \infty}R_t=0$, and if so how can this be proved?

My thoughts: From the definition of a limit and given $R_t\geq 0$, we require that for any $\epsilon >0$, there exists $k \in \mathbf{N}$ such that $R_t<\epsilon\, \forall t>k$. If $R_t$ decreases monotonically it seems obvious that this must hold. However, the fact that the sum is finite does not require $R_t$ to decrease monotonically. It could be that $R_t=0$ for most $t$ and is positive only for occasional $t$ at widely spaced intervals. In that case the result doesn't seem obvious.

Answer 1

$S_N:= \sum^N R_k$ converges, so is a Cauchy sequence, so $|\sum_{k=1}^{n+1}R_k-\sum_{k=1}^{n}R_k|= |R_{n+1}|$ becomes arbitrarily small.

This also shows that the assumption $R_k\ge 0$ can be omitted.