Let $S$ be a non-empty subset of a vector space $V$. The set $\operatorname{span}(S)$ is a subspace of the vector space $V$.
My attempt:
Since $S$ is the subset of $V$, then all vectors that are in $S$ will also be in $V$. Furthermore, any linear combination of the vectors in $S$ will also be in $V$. In other words, $\operatorname{span}(S) \subseteq V$
Need to show closure under vector addition and scalar multiplication
Suppose $S = \{\mathbf{v_{1},v_{2},\cdots,v_{n}}\}$
Then $\operatorname{span}(S) = \{k_{1}\mathbf{v_{1}} + k_{2}\mathbf{v_{2}} \cdots + k_{n}\mathbf{v_{n}} \mid k_1,k_2,\cdots,k_n \in \mathbb R\}$
Consider an arbitrary vector $\bf u$ where $\mathbf{u} \in \operatorname{span(S})$:
$$\mathbf{u} =k_{1}\mathbf{v_{1}} + k_{2}\mathbf{v_{2}} \cdots + k_{n}\mathbf{v_{n}}$$
Multiply by arbitrary scalar, call it $c$:
$$c\mathbf{u} = c \cdot \bigl(k_{1}\mathbf{v_{1}} + k_{2}\mathbf{v_{2}} \cdots + k_{n}\mathbf{v_{n}}\bigr) \implies$$ $$c\mathbf{u} = (ck_{1})\mathbf{v_{1}} + (ck_{2})\mathbf{v_{2}} \cdots + (ck_{n})\mathbf{v_{n}}$$
Since $ck_{1},ck_{2},\cdots ck_{n} \in \mathbb R$, then $c\mathbf{u} \in \operatorname{span}(S)$
Consider one more arbitrary vector, call it $\bf a$ ($\mathbf a \in \operatorname{span}(S))$
$$\mathbf{a} = b_{1}\mathbf{v_{1}} + b_{2}\mathbf{v_{2}} + \cdots + b_{n}\mathbf{v_{n}}$$
We have
$$ \mathbf{u + a} = \sum_{i=1}^{n}k_{i}\mathbf{v_{i}} + \sum_{i=1}^{n}b_{i}\mathbf{v_{i}} = \sum_{i=1}^{n}(k_{i} + b_{i})\mathbf{v_{i}}$$
Since $(k_{1} + b_{1}),(k_{2} + b_{2}), \cdots (k_{n} + b_{n}) \in \mathbb R$, then $\mathbf {u + a} \in \operatorname{span}(S)$
We have closure under vector addition and the scalar multiplication, and hence we can conclude that $S$ is the vector space. And because $S$ is a subset of $V$, then $S$ is a subspace of $V$. $\Box$
Is it correct? I would also like to have a look at better approaches.
Hint: For an arbitrary subset $S$ of $V$, the span is defined as $$\{k_1v_1+\ldots+k_nv_n\mid k_1,\ldots,k_n\in {\Bbb K}, v_1,\ldots,v_n\in S, n\geq 0\}.$$ $\Bbb K$ is the underlying field of the vector space.