I am trying to understand the following proof:
Given $f$ is continuous, prove that for every convergent sequence $(x_n) \to a$ that $\lim_{k\to \infty}f(x_k) = f(a)$
So the prove goes like this (by contradiction and contrapositive?). I found this proof in Spivak's which the whole content will be a quote
If $\lim_{k\to \infty}f(x_k) = f(c)$ were not true, there would be some $\epsilon > 0$ such that for every $\delta$ there is an $x$ with
$0 < |x - a| < \delta$, but $|f(x) - f(a)| > \epsilon$
In particular, for each $n$ there would be a number $x_n$ such that
$0 < |x_n - a| < \delta$, but $|f(x_n) - f(a)| > \epsilon$
Now the sequence $(x_n)$ clearly converges to $c$, but since $|f(x_n) - f(a)| > \epsilon$ for all $n$, the sequence $(f(x_n))$ does not converge to $f(a)$. Which is a contradiction.
When they were doing the contrapositive. They say $\forall \delta >0$, s.t. $\exists \epsilon >0$ etc etc...
Does the epsilon here now depend on $\delta$? If so doesn't that mean it wouldn't make much sense to pick a changing $\delta$? If $\epsilon$ here doesn't change, was the goal just to find some $\delta$ (a decreasing one) as to contradict the non-changing $\epsilon$ ball around $f(x_n)$?