Suppose $\left\{ x_{n}\right\} $ is convergent. Prove that if $c\in\mathbb{R} $, then $\left\{ cx_{n}\right\} $ also converges. Good morning! I wrote a proof of the following exercise but I don't know if it is fine:
Suppose $\left\{ x_{n}\right\} $ is convergent. Prove that if $c\in\mathbb{R} $, then $\left\{ cx_{n}\right\} $ also converges.
Proof:
If $\left\{ x_{n}\right\} $ converges, then by definition
$$\lim_{n\rightarrow\infty}x_{n}=x.$$
For $\frac{\epsilon}{\mid c\mid}>0$, let $N\in\mathbb{\mathbb{N}}$ be such that if $n>N$, then $\mid x_{n}-x\mid<\frac{\epsilon}{\mid c\mid} $.
Now, we have this:
$$|cx_{n}-cx|=|c(x_{n}-x)|=|c|| x_{n}-x|=|c||x_{n}-x|<|c|\frac{\epsilon}{|c|}=\epsilon\Rightarrow c\lim_{n\rightarrow\infty}x_{n}=cx$$ and $\left\{ cx_{n}\right\} $ converges.
Please review this...

That looks fine.  You may want to comment that this only works for $c\neq0$ (for obvious reasons). 
