ThmDex – An index of mathematical definitions, results, and conjectures.

Result R3228 on D65: Cauchy sequence

Bounded sequence need not be Cauchy

Formulation 0

Let $M = (\mathbb{R}, d)$ be the D4369: Standard real metric space such that

(i)	\begin{equation} x : \mathbb{N} \to \mathbb{R}, \quad x_n : = (-1)^n \end{equation}

Then

(1)	$x(\mathbb{N}) \subseteq B(0, 11/10)$
(2)	$x$ is not a D65: Cauchy sequence in $M$

Formulation 1

Let $M = (\mathbb{R}, d)$ be the D4369: Standard real metric space such that

(i)	\begin{equation} x : \mathbb{N} \to \mathbb{R}, \quad x_n : = (-1)^n \end{equation}

Then

(1)	\begin{equation} \forall \, n \in \mathbb{N} : x_n \in (-11/10, 11/10) \end{equation}
(2)	$x$ is not a D65: Cauchy sequence in $M$

Also known as

Counterexample: bounded sequence in a metric space which is not Cauchy

Proofs

Proof 0

Let $M = (\mathbb{R}, d)$ be the D4369: Standard real metric space such that

(i)	\begin{equation} x : \mathbb{N} \to \mathbb{R}, \quad x_n : = (-1)^n \end{equation}

Clear. $\square$