Sequential criterion for functional limits proof in the opposite direction Let $f: A\to\mathbb{R}$ Given $c$ is a cluster point in $A$. Prove that the following statements are equivalent:
(a) The function $f$ does not have a limit at $c$.
(b) There exists a sequence $\{x_n\}$ in $A$ with $x_n \neq c$ for all $n \in \mathbb{N}$ such that the sequence $\{x_n\}$ converges to $c$ but the sequence $\{f(x_n)\}$ does not converge in $\mathbb{R}$
My attempt:
Assume $f$ has no limit at $c$, $x_n$ is a sequence in $A$, $\lim\limits_{n\to \infty}x_n =c$ ie  $\mid x_n-c\mid<\delta$ , WTS $f(x_n)$ does not converge.
If $f(x_n)$ is not bounded, clearly $f(x_n)$ does not converge in $\mathbb{R}$
Otherwise pick a convergent subsequence of $f(x_n)$ let  $L=\lim\limits_{n\to\infty} f(x_n)$ since $\lim\limits_{x\to c}f(x_n)$ does not exist the negation of the definition of a limit says: there exist $\epsilon>0$ such that there is $x \in$  $A\backslash {c}$ with $\mid f(x)-L\mid>\epsilon$
QED
My concern is that the proof is too short and kinda repeats what is given in the question
 A: Let me give you some hints in proving the equivalence of $(a)$ and $(b)$. This is not an easy question if you are new to the subject. First of all, note that it suffices to show that the negation of both statements are equivalent: that is:
(a') $f$ has a limit at $c$, and 
(b') For all $\{x_n\}$ in $A$ such that $\lim_{n\to \infty} x_n = c$, the sequence $\{f(x_n)\}$ converges. 
$(a') \Rightarrow (b')$ is easier: Assume $(a')$ is true. That is, by definition, there is $L$ so that:
(*) For all $\epsilon >0$, there is $\delta >0$ so that $|f(x) - L| < \epsilon$ whenever $x\in A$ and $|x-c| < \delta$. 
Now let $\{x_n\}$ be a sequence in $A$ converging to $c$. Then for all $\delta >0$, there is $N_\delta \in \mathbb N$ so that $|x_n - c|< \delta$ whenever $n \geq N_\delta$. Try to use this and (*) to show 
$$\lim_{n\to \infty} f(x_n) = L.$$
For $(b') \Rightarrow (a')$, you might try to proof the following two facts: Assuming $(b')$, then
(i) For any $\{x_n\}$ in $A$ and $lim_{n\to \infty} x_n = c$, the sequence $\{f(x_n)\}$ converges to the same real number $L$ (independent of sequence $\{x_n\}$ chosen. 
(ii) Use (i) to show that $\lim_{x\to c} f(x) = L$. 
For both (i) and (ii), it will be easier to argue by contradiction. 
