Let $(V, \| \cdot \|)$ be a normed space, and $S \subset V$ a vector subspace. Then $\overline{S}$ is a vector subspace in $V$

Proof

First of all, $0 \in \overline{S}$ because $0 \in S$ Now, let $x, y \in \overline{S}$ and $\lambda \in K$ (where $K$ is the ground field of the vector space $V$ . Then there are two sequences in $S$ say $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ which converge to $x$ and $y$ respectively.

Then, the sequence $(x_n + \lambda \cdot y_n)_{n \in \mathbb{N}}$ is a sequence in $S$ (because $S$ is a vector subspace), and it's trivial (use properties of the norm) that this sequence converges to $x + \lambda \cdot y$ and so this sum is a vector which lies in $\overline{S}$

We have proved that $\overline{S}$ is a vector subspace. QED.