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Baby Rudin Notes

Priciple of Mathematical Analysis

Chapter 1: The Real and Complex Number Systems

1.10 Definition: Least-upper-bound property

If $E\subset S$ and $E$ is bounded above, then $supE$ exists in $S$.

1.11 Theorem

S is a ordered list set with least-upper-bound property. Then, $\mathrm{\forall }E\subset S$, if $E$ is nonempty and bounded above, then $supE$ exists in S.

Proof

1.19 Theorem

There exists an ordered field $\mathbb{R}$ which has the least-upper-bound property.

Proof:

1.20 Theorem

(a)[archimedean property] If $x\in \mathbb{R},y\in \mathbb{R},\text{ and }x>0$, then there exists a positive integer $n$ such that $nx>y$.
(b) If $x\in \mathbb{R},y\in \mathbb{R},\text{ and }x<y$, then there exists a rational number $p$ such that $x<p<y$.

Proof:

Chapter 2: Basic Topology