I want to prove: $\{c_n\}$ converges to $c$ if and only if $\{c_n -c\}$ converges to $0$
so I start with ( $\{c_n\}$ converges to $c$ ) $\Rightarrow$ ( $\{c_n -c\}$ converges to $0$ ) and I assume $\{c_n\}$ converges to $c$.
So then $$\lim_{n\to\infty}\{c_n -c\} =\lim_{n\to\infty}\{c_n\} - \lim_{n\to\infty}\{c\} $$ since $$\lim_{n\to\infty}\{c_n\} = c $$ by assumption and $c$ is a constant so $\lim_{n\to\infty}\{c\} = c$ the limit of $$\lim_{n\to\infty}\{c_n -c\} = c- c = 0$$
next I go to ( $\{c_n -c\}$ converges to $0$) $\Rightarrow$ ( $\{c_n\}$ converges to $c$) and I assume $\{c_n -c\}$ converges to $0$
so $$\lim_{n\to\infty}\{c_n -c\} =\lim_{n\to\infty}\{c_n\} - \lim_{n\to\infty}\{c\} = 0$$ and again $c$ is a constant so $$\lim_{n\to\infty}\{c\} = c$$
thus $$\lim_{n\to\infty}\{c_n -c\} =\lim_{n\to\infty}\{c_n\} - c = 0$$ add $c$ to both sides and I get $$\lim_{n\to\infty}\{c_n\} = c$$
I think this is basically correct. I show both directions of the if and only if. Can anybody see any mistakes in my reasoning? I have a Theorem from the textbook for the separation of the limit into two limits and I'm allowed to say the limit of a constant is that constant directly. I know nothing else about $\{c_n\}$ and $c$ is an arbitrary number in $\mathbb{R}$.