Help with convergence proof: ${c_n}$ converges to c if and only if ${c_n -c}$ converges to 0 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 $\mathbb{R}$.
 A: It's basically correct, but I think it can be improved.  It seems that you are assuming in the second part that $\lim\limits_{n\rightarrow\infty} c_n$ exists at the outset. But you can't do this.
What you do is start with two limits that you know exist. Then you can say the limit of the sum exists and is the sum of the limits.
More formally:

If $\lim\limits_{n\rightarrow\infty} c_n=L$ and if
  $\lim\limits_{n\rightarrow\infty} d_n=M$, then
  $\lim\limits_{n\rightarrow\infty} (c_n+d_n)$ exists and is equal to
  $M+L$: $$
 \lim\limits_{n\rightarrow\infty}(c_n+d_n) =
 \lim\limits_{n\rightarrow\infty} c_n+\lim\limits_{n\rightarrow\infty}
  d_n=L+M. $$

So I would do the second part as follows:
We know that
$\lim\limits_{n\rightarrow\infty}(c_n-c)=0$ and that $\lim\limits_{n\rightarrow\infty}   c = c$. 
Then,
$\lim\limits_{n\rightarrow\infty}\bigl ( (c_n-c)+c\bigr)  $ exists. But, $(c_n-c)+c=c_n$; so $\lim\limits_{n\rightarrow\infty} c_n$ exists and
$$
\lim\limits_{n\rightarrow\infty} c_n = \lim\limits_{n\rightarrow\infty} \bigl ((c_n-c)+c\bigr)= \lim\limits_{n\rightarrow\infty} (c_n-c) +\lim\limits_{n\rightarrow\infty} c =0+c=c.
$$
