A doubt in the rigorous definition of limits. I studied the definition of limits today, and I think I mostly understood it, but I have a little doubt. In the definition:
$f(x)$ is defined on some open interval containing $a$, except at possibly $a$. So, $\lim_{x\to a} f(x) = L $ if and only if for every number $\varepsilon>0$, there exists a corresponding number $\delta>0$ such that: 
$$\text{If   } 0<|x-a|<\delta \text{ then } |f(x)-L|<\varepsilon$$
First of all, if you see any mistake in my definition or understanding, please tell. So my question is, why can't we write $0<|x-a|\leq\delta$ and $|f(x)-L|\leq\varepsilon$ in the last line? What kinds of problems can arise from this? If the definitions are equivalent, can you explain/prove me why?

Here is a very, very quick answer that should help you:


*

*if you want $\leq \varepsilon$, just choose $2\varepsilon$ in the definition with $<$;

*if you want $\leq \delta$, just choose $\frac{\delta}{2}$ in the definition with $<$.


In the first case, you exploit the words "for all", in the second the words "for some".
