I am learning $\epsilon$-$\delta$ definition of limits. I was confused on a few points and read some of the related answers on this and other sites. But I couldn't find discussion on any of these questions anywhere, so I am asking them here. The questions are-
Why should $|x-c|<\delta$ imply $|f(x)-L|<\epsilon$ ? Can't it be the other way round? That is, would this be wrong definition:
" Let $f$ be a function defined on an open interval around $c$ (except possibly at $c$). Then, if for any $\epsilon>0$ there exists a $\delta > 0$ such that $0< |x-c|<\delta$ whenever $|f(x)-L|<\epsilon$; then $$\lim_{x \to c} f(x) = L$$ "?
2 Why do we choose open intervals around $x$ and $f(x)$ instead of closed ones (i.e. $x \in (x-\delta,x+\delta),f(x) \in (f(x)-\epsilon,f(x)+\epsilon)$ instead of $x \in [x-\delta,x+\delta],f(x) \in [L-\epsilon,L+\epsilon]$ ) ?
- Why are $\delta$ and $\epsilon$ chosen to be greater than zero? Can't we choose them to be less than zero and specify the bounds of $x$ and $f(x)$ thus: $\delta < x-c< -\delta$ and $\epsilon < f(x) - L < -\epsilon$ ?
My guess is that the answer to the points 2 & 3 should be: It is just by convention. But I have no source to back it up. I couldn't find any discussion on this anywhere. Not in my textbook, nor on Wikipedia or other sites.
So, what are the answers to these questions?
EDIT: If someone says that we can make the choices I suggested in points 2 & 3 too (instead of the ones we do currently), and so the latter are just historical conventions, please cite the source for your statement.