Show, using only the definition, that if $\{X_n\}$ is Cauchy and $C\in\mathbb{R}$ then $\{CX_n\}$ is Cauchy.

Question

Let me know if what I did this correct please.

Let $\epsilon>0$ be given, we want to find $N\in\mathbb{N}$ such that $|CX_n-CX_m |<\epsilon$ $\forall n,m\geq N$. $$|CX_n-CX_m |=|C||X_n-X_m |$$ $∃N\in\mathbb{N}$ such that $|X_n-X_m |<\epsilon/|C|$ $\forall n,m\geq N$ Hence $CX_n$ is Cauchy.