For writing up the $epsilon-delta$ proof, what is the order of $|f(x)-L|$ and $|x-a|$ at the end?

To elaborate, do you...

1. $0<|x-a|<\delta$ and then add to that so that you build up to $|f(x)-L|$ and then show that this is less than $\epsilon$ by using the $\epsilon-\delta$ relationship you found earlier.


2. $|f(x)-L|$ and then break it down so you can substitute $|x-a|<\delta$ and then show that the former is less than $\epsilon$ by using the $\epsilon-\delta$ relationship you found earlier.

In the way the final part of the definition is written ("if $0<|x-a|<\delta$, then $|f(x)-L|<\epsilon$"), does it matter which choice you use, or it just preference? I've seen books do both ways, but from what I've seen, the second option is more common.

In terms of "If P, then Q" which would you want to work with first - P, or Q?