Question about lemma (2) - Spivak' calculus - page 89

Question about lemma (2) - Spivak' calculus - page 89

I have a question about the proof of lemma (2) in Spivak's calculus, page
89. How does he simplifies $$ \ |y_0|\frac{\epsilon}{2(|y_0|+1)} $$ to get
$$ \ \frac{\epsilon}2 $$ Thanks.