Proof of Simple Limit Theorem

Question

Give epsilon N type proof of the SLT. Namely, if $\{a_n\}$ converges to $A$ and if $c$ is a real number then $\{ca_n\}$ converges to $cA$.

so far I have that for every $\varepsilon>0$, $|ca_n-cA|<\varepsilon$ which can then be rewritten as $|c||a_n-A| < \varepsilon$. so if $c=0$ then it holds true. but i dont know how to go about saying its true when $c\not= 0$

Answer 1

Let $\varepsilon >0$. Choose $N$ s.t. for all $n>N$, $|a_n-A|<\dfrac{\varepsilon}{|c|}$.

Now for $n>N$, we have $|ca_n-cA|=|c||a_n-A|\leq|c|\dfrac{\varepsilon}{|c|}=\varepsilon$