Show that a subset $W$ of a vector space $V$ is a subspace of $V$ iff $\text{span}(W) = W$.
MY ATTEMPT
We may consider that $W\neq\varnothing$. Otherwise $\text{span}(W) = \{0\} = \varnothing$ which makes no sense.
Suppose that $W\leq V$. Then $x + y \in W$ and $ax\in W$ whenever $x,y\in W$ and $a\in\textbf{F}$.
Consequently, if $w\in\text{span}(W)$, then we have that \begin{align*} w = a_{1}x_{1} + a_{2}x_{2} + \ldots + a_{n}x_{n} \in \text{span}(W) \Rightarrow w\in W \end{align*}
On the other hand, if $w\in W$, then $1w\in\text{span}(W)$, and we are done.
Conversely, let us suppose that $\text{span}(W) = W$.
Then, if we are given $x,y\in W$ and $a\in\textbf{F}$, we have that $x + y \in\text{span}(W) = W$ and $ax\in\text{span}(W) = W$.
Thus we conclude that $W$ is a linear subspace of $V$ and we are done.
Any comments on my solution are welcome.