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) $x(\mathbb{N}) \subseteq B(0, 2)$
(2) $x$ is not a D65: Cauchy sequence in $M$
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. $\square$