Is the proof of $\lim ca_n=ca$ correct using the definition of the limit

Question

Using the definition of the limit of a sequence, Prove that if $\lim a_n=a$, then $\lim ca_n=ca, \forall c \in \mathbb{R}$.

My attempt:

Case $I$:If $c=0$, The proof is trivial.

Case $II:$ If $c \ne 0$, We choose $\epsilon_1=\frac{\epsilon}{|c|}$, where $\epsilon_1,\epsilon>0$.

Now since $\lim a_n=a$, $\exists n_0 \in \mathbb{N} $ such that $$|a_n-a|<\epsilon_1,\forall n \geq n_0$$

Multiplying with $|c|$ both sides we get
$$|ca_n-ca|<\epsilon, \forall n \geq n_0 \Rightarrow \lim ca_n=ca$$

Answer 1

This proof looks fine to me, but don’t forget to add additional steps that you have used. I would write: $$ |ca_n-ca|=|c||a_n-a|<|c|\epsilon_1=\epsilon $$