If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a function that is not continuus at $c$, show that there exists a sequence $(x_n)$ such that $\lim_{n \to \infty}x_n=c$, but such that $f(x_n)$ does not converge to $f(c)$.
My Solution;
Firstly if $f$ is not continuous at $c$ then;
$$\exists \epsilon>0 \ \ s.t \ \ \forall \delta>0 \ \ \exists x \in \mathbb{R} \ \ s.t \ \ |x-c|<\delta \ \ and \ \ |f(x)-f(c)|\geq \epsilon $$
Now if we define a convergent sequence $x_n$ ;
$$ \exists \ N_0 \in \mathbb{N} \ \ s.t \ \ \forall n>N_0 \ \ |x_n-c|<\dfrac{1}{n}$$
Then by these definitions if we let $\delta=\dfrac{1}{n}$ then $\lim_{n \to \infty}x_n=c$ and $f(x_n)$ does not converge to $f(c)$
Is this solution correct? Any feedback would be appreciated