I am stuck with trying to prove the following:
$\lim_{x\rightarrow a}f(x)=l$
Prove that $\lim_{n\to\infty}f(x_n)=l$ for every sequence $(x_n)^\infty_{n=1}$ such that $\lim_{n\rightarrow \infty}x_n = a$
I'm really not sure how to go about doing that.
Thanks for your time.
$\lim_{x\rightarrow a}f(x)=l$ means:
$(*)$ for every $\epsilon >0$ there is $ \delta >0$ such that $|x-a|< \delta$ implies $|f(x)-l|< \epsilon$.
Now let $(x_n)^\infty_{n=1}$ be a sequence with $x_n \to a$. We have to show that $f(x_n) \to l$.
To this end let $ \epsilon >0$. Now choose $\delta$ as in $(*)$. There is $N \in \mathbb N$ such that $|x_n-a|< \delta$ for all $n>N$.
Hence $|f(x_n)-l|< \epsilon$ for all $n>N$.