The theorem in my textbook says Assume $A \subseteq \mathbb{R}$, $f:A \rightarrow \mathbb{R}$, and $c$ is a limit point of A. Then $\lim_{x\to\ c} f(x) = L$ iff for every sequence $a_n$ from $A$ for which each $a_n \neq c$ and $a_n \rightarrow c$ we have $f(a_n) \rightarrow L$.
Now I am confused about the proof of the reverse direction. The textbook does this by contradiction. That is, they assume that for every sequence $a_n \rightarrow c$ that $f(a_n) \rightarrow L$, but $\lim_{x\to\ c} f(x) \neq L$.
This makes sense so far.
So next, they state $\lim_{x\to\ c} f(x) \neq L$ in terms of epsilons and deltas. That is, they say: there exists $ε>0$ such that for all $δ>0$ there exists some $x \in A$ satisfying $0<|x-c|<δ$ for which $|f(x)-L| \geq ε$.
This makes sense.
Now they say, in particular, for this $ε>0$, by letting $δ_n=1/n$ we see that there exists some $x_n \in A$ satisfying $0<|x_n-c|<1/n$ for which $|f(x_n)-L| \geq ε$. Then they note that $|x_n-c|<1/n$ implies $x_n \rightarrow c$, so they say they've found a contradiction.
This is where I'm confused... is $x_n$ a sequence of numbers from A? Also why is there even a subscript under the $δ$ ... This notation is making no sense to me. I understand that we are trying to find a sequence of numbers $a_n$ that converge to c for which $f(a_n)$ does not converge to L, but I don't see how this does that.