Given $S_n=a_1+a_2+a_3+\cdots+a_n$, both $\lim\limits_{n\to\infty}{a_n}$ and $\lim\limits_{n\to\infty}{S_n}$ exists.
Is the following equation correct? If not, give a counter example please.
$\lim\limits_{n\to\infty}S_n=\lim\limits_{n\to\infty}{a_1}+\lim\limits_{n\to\infty}{a_2}+\lim\limits_{n\to\infty}{a_3}+\cdots+\lim\limits_{n\to\infty}{a_n}$