"Definition: For every $\epsilon > 0$ there is some $\delta > 0$ such that, for all $x$, if $0 < |x - a| < \delta$, then $|f(x) - l| < \epsilon$ This is if $f$ approaches the limit $l$ near $a$. "
It says for some $\delta$ so all we need to do is prove there is only one $\delta$ such that $|f(x) - l| < \epsilon$ which could mean $\delta = \epsilon$ or we could just choose one delta right?
Thanks!
Semantically, you have the right idea. "...There is some $\delta > 0$" means that only one suffices to satisfy the definition.
But don't forget the first part of the statement: "For all $\epsilon > 0$..." It is very likely that the $\delta$ you supply depends on $\epsilon$. In some cases it may suffice to let $\delta = \epsilon$, but certainly not all.