Definition (As written in Michael Spivak's Calculus)
The function $f$ approaches a limit $l$ near $a$ means: for every $\epsilon >0$ there is some $\delta > 0$ such that, for all $x$, if $0<|x-a|<\delta$, then $|f(x)-l|<\epsilon$.
my question is: why can't it be: $$0<|x-a|\leq \delta,|f(x)-l|\leq \epsilon$$ After looking at limits of functions for a long time just to grasp it's meaning and using the definition quite a lot solving homework I realized I keep writing the same inequality without really understanding why.
The only explanation given in Spivak's book for this part of the definition goes over it without explaining the inequality. I tried looking for an explanation myself but wasn't really able to find anything wrong with it. Is it also possible to write the definition like that or is there a problem with that?
(first non-homework related question :p)