First, this is lambda: . And this is delta: .

As for your question: in this kind of proof, we need to be small, and sometimes we need to be smaller than, for instance, [tex]\delta_1[/tex] and . Then this is equivalent to .

This is just taking small enough so that and . If you know that something is true if [tex]|x-5|\leq \delta_1[/tex] and , then this is true if .