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.

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.

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.

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October 17, 2026 1 min read

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