Let $X$ be a metric space, $E$ be a subset of a metric space $X$, $p$ be a limit point of $E$, $f:E \to \mathbb{C}$, and $\lim\limits_{x \to p} f(x)=L$. Finally let $c \in \mathbb{C}$. Show that $\lim\limits_{x \to p} cf(x)=cL$.
Let $\epsilon > 0$ be given. Want to show that there exists $\delta>0$ such that $|cf(x)-cL|<\epsilon$ whenever $|x-p|<\delta$.
Suppose $c\neq 0$.
Then $|cf(x)-cL|< \epsilon \Rightarrow |c||f(x)-L|<\epsilon \Rightarrow |f(x)-L|<\frac{\epsilon}{|c|}$.
I'm stuck on showing that $\delta>0$ exists. Or, I may be potentially doing this proof wrong.