# 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 $q \in B$.