Quantifiers for formal definition of a limit

Question

I have a problem. I have to prove that

$(\forall\,\epsilon > 0)(\exists\,\delta > 0)[0 < |x − a| < \delta \implies |f(x) − L| < \epsilon$ (formal definition of a limit)

$(\exists\,\delta > 0) (\forall\,\epsilon > 0)[0 < |x − a| < \delta \implies |f(x) − L| < \epsilon$

are the same. Basically the only difference is the order of the quantifiers.

My first instinct was to use DeMorgans Law and negate the quantifier but that is not quite the same proposition. I am not sure how to prove this equivalence. Can anybody help?

Answer 1

$\forall \epsilon \;\;\exists \delta$ means for any given $\epsilon$ we have $\delta$ which depends on $\epsilon$.

$\exists \delta\;\;\forall \epsilon $ means there is a $\delta$ for all $\epsilon$. In this case $\delta$ is independent of $\epsilon$

So your statements are NOT same.

Answer 2

Another consequence of $(\exists\delta>0)(\forall\epsilon>0)(|x-a|<\delta\implies |f(x)-L|<\epsilon)$. Since $|f(x)-L|<\epsilon$ for all $\epsilon>0$, we must have $f(x)=L$ whenever $|x-a|<\delta$.