Hi, On page 150-151 of 3rd ed, before lemma 7.23, it says that under a stronger assumption that the function F is an open mapping, Lemma7.23 can be proved without using Brouwer fixed point theorem. I can't finish the proof avoiding Brouwer fixed point theorem. What I have done is prove that either F(B) is including B(0,1-epsilon) or F(B) is disjoint with B(0,1-epsilon), however, I cann't exclude the case that they are disjoint. Someone helps me? I have attached the screenshot of the paragraph involved. Thanks in advance!