# Completeness of the Real Numbers¶

## Least upper bound property¶

Every nonempty subset of \(\mathbb{R}\) having an upper bound has a least upper bound.

## Monotone convergence theorem¶

If a sequence of \(\mathbb{R}\) is monotone and bounded, then it converges.

## Nested interval theorem¶

For a sequence of **closed** intervals \(\{[a_n,b_n]\}\), if \([a_{n+1},b_{n+1}]\subset[a_n,b_n]\) for all \(n\in\mathbb{N}^\star\) and \(\lim\limits_{n\to\infty}{|a_n-b_n|}=0\), then exists exactly one \(\xi\in\mathbb{R}\) so that \(\xi\in[a_n,b_n]\) for all \(n\in\mathbb{N}^\star\).

## Bolzano-Weierstrass theorem¶

Every bounded sequence of \(\mathbb{R}\) has a convergent subsequence.

## Cauchy criterion¶

A sequence of \(\mathbb{R}\) is convergent if and only if it is a fundamental sequence.

Fundamental sequence: A sequence \(\{x_n\}\) is called a fundamental sequence if for all \(\varepsilon > 0\), exists \(N\in\mathbb{N}^\star\) so that \(|x_{n+p} - x_n| < \varepsilon\) for all \(n>N\) and \(p\in\mathbb{N}^\star\). Fundamental sequence is also known as Cauchy sequence.

## Heine-Borel theorem¶

If \(H\) is an infinite **open** cover of **closed** interval \([a, b]\), then exists a finite subset \(H_0\subset H\) so that \(H_0\) is an **open** cover of \([a,b]\).

Or: If \(H = \{(a_\alpha, b_\alpha) | \alpha \in \Lambda\}\) which satisfies \(\bigcup\limits_{\alpha\in\Lambda}{(a_\alpha, b_\alpha)} \supset [a, b]\), then exists a finite subset \(H_0\subset H\) so that \(\bigcup\limits_{(a_\alpha, b_\alpha)\in H_0}{(a_\alpha, b_\alpha)} \supset [a, b]\).

Created: 2023-10-16