Cauchy Sequence

A sequence $u:\mathbb{Z}^+ \rightarrow A$ is Cauchy iff: $$

$$\forall \epsilon \gt 0\, \exists N \in \mathbb{Z}^+\, \forall m,n; m,n \gt N \implies \lvert u_n - u_m \rvert \lt \epsilon$$

Bounded

$$u_n\;\text{is Cauchy} \implies u_n\;\text{is bounded}$$

Converse is not true.

Proof Hint

• Consider the Cauchy definition
• Take $n \gt m = N + 1 \gt N$

Convergence & Cauchy

$$u_n\;\text{is converging} \implies u_n\;\text{is a Cauchy sequence}$$

Converse is true only when the sequence is a subset of a Complete set.

Proof Hint

• Consider the limit definition of converging sequences
• Introduce the converging value (say $L$) into the inequality and split into 2 parts

Complete

A set $A$ is complete iff: $$

$$\forall u:\mathbb{Z}^+ \rightarrow A;\; u\;\text{converges to}\; L \in A$$

IMPORTANT: $\mathbb{R}$ is complete. Proof is quite hard. $$

IMPORTANT: $Q$ is not complete because: $$

$$\sum_{k=1}^\infty \frac{1}{k!} = e - 1 \not\in \mathbb{Q}$$

IMPORTANT: $\mathbb{Z}$ is complete. $$