Proof explanation: $\sum_{k=1}^{\infty}{a_k}<\infty \implies a_k \overset{k\to \infty}\longrightarrow 0$

Question

Proof explanation: $\sum_{k=1}^{\infty}{a_k}<\infty \implies a_k > \overset{k\to \infty}\longrightarrow 0$

Here's the proof:

$a_k:=s_k-s_{k-1}$. Since $s_k \to s$, we know that $s_{k-1}\to s$ and thus we can conclude that $a_k \to s-s=0$ for $k\to \infty$.

$s_k$ is the partial sums.

What I don't understand in this proof is $a_k:=s_k-s_{k-1}$.... ?

Answer 1

$$s_k-s_{k-1}=(a_k+a_{k-1}+...+a_2+a_1)-(a_{k-1}+a_{k-2}+...+a_2+a_1)=a_k$$ All other summands cancel out.