Everything that follows below is under the context we are working on normed vector spaces.
Some context. When trying to prove a subspace $S \subset X = (X,\| \cdot\|_X)$ is closed, the first intuition would be to show that $S = \overline{S}$, where $\overline{S} = S \cup S'.$ The inclusion $S \subset \overline{S}$ comes simply by definition of $\overline{S}.$ Normally, the only thing one has to do is to show that $\overline{S} \subset S$ and thus the subspace $S$ is closed.
To do so, we pick an arbitrary element of $\overline{S}$, say $s \in \overline{S}$ and certainly there is a convergent sequence $(s_k)_{k \in \Bbb N} \subset S$ such that $s_k \rightarrow s$, as $k \rightarrow \infty$.
It is also well known that every convergent sequence is also a Cauchy sequence and here urges my question. We know that the sequence $(s_k)_{k \in \Bbb N}$ is a sequence where each element is in the subspace $S$ and at the same time it converges to an element $s \in \overline{S}.$
Can we say $(s_k)_ {k\in \Bbb N}$ is a cauchy sequence in $S$ or/and can we say that $(s_k)_{k \in \Bbb N}$ is a cauchy sequence in $\overline{S}$? If so, why?
In other words, if a sequence is defined in a set $S$ but converges to a point not necessarily in $S$ is it a Cauchy sequence on $S$?
Thanks for any help in advance.