double limit of a series summation

Question

There is a sequence $a_n$ and $a_n \to 0$ as $n \to \infty$. Let $S_N = \sum_{n=0}^{N} a_n$. Let the sequence $S_N$ be converging to a limit $l$ where $l \in \mathbb{R}$. What is $\lim_{K\to\infty}\lim_{N\to\infty}\sum_{n=K}^{N} a_n $ ?

Answer 1

0.

$l = \lim_{N\to\infty}\sum_{n=0}^{N} a_n$. Therefore $\lim_{N\to\infty}\sum_{n=K}^{N} a_n = l - \sum_{n=0}^{K-1} a_n = l - S_{K-1}$.

But since the sequence converges to $l$, $\lim_{K\to\infty} |S_{K} - l| = 0$.