TITLE: Proof: Series converges $\implies $ the limit of the sequence is zero
QUESTION [11 upvotes]: I've been using the sentence:

If a series converges then the limit of the sequence is zero

as a criterion to prove that a series diverges (when $\lim \neq 0$) and I can understand the rationale behind it, but I can't find a formal proof.
Can you help me?

REPLY [13 votes]: Yes.
$$\lim_{n \to \infty} \left ( \sum_{k = 1}^{n + 1} a_k - \sum_{k = 1}^{n} a_k \right ) = \lim_{n \to \infty} a_{n + 1} $$
And both sums will converge to the same number so the limit is zero. This is by far the easiest proof I know.
This is the Cauchy criterion in disguise by the way, so you could use that too.

REPLY [3 votes]: If we know that the sequence converges and merely wish to show it converges to zero, then a proof by contradiction gives a little more intuition here (although the direct proofs are simple and beautiful). Assume $a_n\to a$ with $a>0$, then for all $n>N$ for some large enough $N$ we have $a_n > a/2$ (take $\varepsilon = a/2$ in the definition of the limit). Now the sum diverges: $\sum_{n>N}a_n > \sum_{n>N}a/2 = \infty$. A similar argument works when $a<0$.