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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!

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# A question on the paragraph before 7.23 of Rudin's Real and complex analysis

Can you offer guidance or do you also need help?

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