Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $(X,d)$ be a metric space, with open balls $B_{d}(x,r)$ and the topology $\mathcal{T}_{d}$ of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces §topology. Let $A\subseteq X$, and let $d_{A}=d|_{A\times A}$ be the restriction of $d$ to $A\times A$; it is a map $d_{A}:A\times A\to\mathbb{R}$ with $d_{A}(a,b)=d(a,b)$ for all $a,b\in A$, by Domains and Restrictions of Maps §restriction-map and Domains and Restrictions of Maps §restriction-value. Sequences are as in Sequences §sequence; convergence to a point, convergent sequences, Cauchy sequences and completeness are as in Convergent Sequence in a Metric Space §converges, Convergent Sequence in a Metric Space §convergent, Cauchy Sequence in a Metric Space §cauchy and Complete Metric Space §complete, and closed subsets as in Closed Subset of a Topological Space §closed.

¶ $d_{A}$ is a metric on $A$, so that $(A,d_{A})$ is a metric space.

¶ Let $(a_{k})$ be a sequence in $A$; it is also a sequence in $X$. For every $a\in A$, $(a_{k})$ converges to $a$ in $(A,d_{A})$ if and only if it converges to $a$ in $(X,d)$. Moreover, $(a_{k})$ is a Cauchy sequence in $(A,d_{A})$ if and only if it is a Cauchy sequence in $(X,d)$.

¶ If $A$ is closed in $(X,\mathcal{T}_{d})$, $(a_{k})$ is a sequence in $A$, $x\in X$, and $(a_{k})$ converges to $x$ in $(X,d)$, then $x\in A$.

¶ If $(X,d)$ is complete and $A$ is closed in $(X,\mathcal{T}_{d})$, then $(A,d_{A})$ is complete.