I apologize in advance if this turns out to be a trivial question, but when we define the limit of a sequence $x_n$ (it it exists) as the number $a$ such that $\forall \delta > 0\ \exists r : n \ge r \Rightarrow | x_n - a | < \delta$, is there a particular reason why we use $< \delta $ instead of $\le\ \delta$?
I first thought of this when reading the proof of the uniqueness of the limit. And indeed that proof does not hold if we use the "less or equal" form. But then I failed at trying to construct a sequence that under that new definition, would have two distinct limits. So I'm thinking that must not be the reason. So why is it that the definition uses the $< \delta $ form? Is it just a matter of convention?
Thanks in advance.