In the case where is a number, then you would prove by showing that . What this translates to is that as you make approach and reduce the distance between them, then approaches and the distance between them also gets smaller.
But when you want to make , you can't make "close in" on a value. So instead we say that no matter how big we make , we will still have close in on . To do this, provided we choose some very large value for , say , if we can show that any value greater than this will still make approach , then we prove .