limit of a closed convergent subset is in the subset

Question

Let S be a closed subset of C. Show that for any convergent sequence of elements of S, the limit of the sequence also belongs to S.

I know how to prove the subset is closed given the other conditions, i was wondering if i can prove this statement directly or by a proof by the contrapositive? I cant see how to directly prove this

Answer 1

Let $x_1,x_2,\ldots$ be a convergent sequence in $S\subseteq\mathbb{C}$ with limit $x\in\mathbb{C}$. Then for each $\epsilon>0$ there is an $N_\epsilon\in\mathbb{N}$ such that $|x-x_n|<\epsilon$ for each $n\geq N_\epsilon$. Thus each $\epsilon$-ball around $x$ contains some element of $S$, namely $x_{N_\epsilon}$, hence $x$ belongs to the closure of $S$. Since $S$ is closed, it follows that $x\in S$.