Let $f$ be a bounded real-valued function on a subset of $\mathbb{R}$ and let $x_{0} \in \mathbb{R}$ be a cluster point with respect to $A$. Suppose that $\lim_{x\to x_0} f(x)$ does not exist. Show that there exist two sequences $(x'_n)$, $(x''_n)$ converging to $x_0$ with $\lim_n f(x'_n)=l'$ and $\lim_n f(x_n'')=l''$ such that $l' \not= l''$.
My attempt:
Since $\lim_n f(x_n')$ does not exist, using Cauchy convergence criterion, $\exists \epsilon >0, \,\forall \delta>0$ such that when $|x_n-\delta|<0$ and $|x_m-\delta|<0$, $|f(x_n)-f(x_m)|\geq \epsilon$.
And then I do not know how to proceed and I also do not know whether my beginning steps in proof are correct or not. Could anyone kindly help?An $\epsilon-\delta$ argument is preferred.
Since $f$ is bounded on $A$ it follows that $$\limsup_{x\to x_0}f(x)=l',\liminf_{x\to x_0} f(x) =l''$$ exist. If they are equal then it is a well known theorem that the limit of $f$ exists. By given condition thus $l' \neq l''$.
Further by definition of limsup, liminf there are sequences $x_n' \to x_0,x_n''\to x_0$ such that $f(x_n') \to l', f(x_n'') \to l''$.