I have a question about the last inequality Rudin uses in his proof of this theorem. Given that |z| < 1 he gets the inequality(adsbygoogle = window.adsbygoogle || []).push({});

|(1-z^(m+1)) / (1-z)| <= 2 / (1-z)

I think he is using the fact that |z| = 1, so

|(1-z^(m+1)) / (1-z)| <= (1 + |z^(m+1)|) / |1-z|

So i am guessing that

|z^(m+1)| < 1 since |z| < 1

But I don't know why this would be true?

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# Rudin theorem 3.44

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