so the idea is that given any epsilon, you can select a delta that will satisfy the inequality.
the tricky part about these proofs are that you are essentially working backwards.
So, in this example, you "start" by saying,
Then we have to prove that
so to prove P implies Q, start by assuming P and show that Q holds.
so assume that
(we got to "pick" )
then we're done.
so by the definition of limit,
for each there is a (in fact, )
then you can say
Hope that may make things clearer....
What is the point of epsilon/delta? Can't you just prove this using right/left hand limits?
Limits are DEFINED by the epsilon/delta definition. That is to say, whenever we take limits we are using the definition implicitly, and in the teaching of calculus, before limits are defined this way, the limit is not defined at all. We can only say things like "f(x) gets really really close to L as x gets closer and closer to a" which isn't very formal, so this is essentially a formalization of the above statement.
Hopefully everything I've said it correct and helps you understand a bit!