Research Polymath

July 31, 2009

Principles of Mathematical Analysis by Walter Rudin

Filed under: Analysis — ramanujantao @ 4:56 pm
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Chapter 1: The Real and Complex Number Systems

One thing I notice about Rudin in general is that he tends to “pull rabbits out of hats.” For instance, on pg.2, he does the following: Let A = \{p: p^2<2, \ p \in \mathbb{Q}, \ p>0 \} and B = \{p: p^2>2, \ p \in \mathbb{Q}, \ p>0 \}. He wants to show that A contains no largest element and B contains no smallest. Then all of a sudden he does the following: For every p>0 associate q = p-\frac{p^2-2}{p+2} = \frac{2p+2}{p+2}. Then consider q^2-2 to show that q \in A or q \in B.

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