I mean, if $S\subset W$ and $W$ is a subspace, then $S$ is either a basis for $W$ or at least spans some subset of $W$, therefore $\operatorname{span}(S)\subseteq W$.
Is it ok? For finite sets it's okay to me, but I'm talking about infinite sets.
Span:
Suppose a vector space $(V,+,\cdot)$, and
$$S = \{u_1,\cdots,u_n\}$$
(and $S$ is a subset of $V$, not a subspace)
$$[S]=:\cap_{w\subset V, w\supseteq S} W$$
In other words, $[S]$ is, by definition, the intersection of all $W$, such that $W$ is a subspace of $V$ and $W$ contains $S$.
Update: in fact, using this assumption, I have:
$$[S_1]\subseteq W$$ $$[S_2]\subseteq W$$
Can I prove that $$[S_1]+[S_2]\subseteq W$$ too? *where $+$ means sum of subspaces
It is ok, but for a proof, I need your definition of span$(S)$.
Edit:
If $S\subset W$, then the intersection of all subspaces that contain $S$ certainly is contained in $W$, because $W$ is one of such subspaces. Note that we haven't used that $S$ or $\dim W$ are finite.
Proof of $[S_1]+[S_2]=[S_1\cup S_2]$. Indeed: if $v\in[S_1\cup S_2]$ then $v$ is in every subspace that contains $S_1\cup S_2$, and $[S_1]+[S_2]$ is certainly one of such subspaces, so $v\in[S_1]+[S_2]$. Conversely, if $v\in[S_1]+[S_2]$ then there are some $w_i\in[S_i]$ (for each $i\in\{1,2\}$) such that $v=w_1+w_2$, which is in $[S_1\cup S_2]$.
Indeed, $w_i$ is in every subspace that contains $S_i$. Take a subspace $U$ that contains $S_1\cup S_2$. Since $U$ contains $S_1$, $w_1\in U$, and since $U$ contains $S_2$, $w_2\in U$. Then, $w_1+w_2\in U$. Therefore, $w_1+w_2$ is in every subspace that contains $S_1\cup S_2$, and thus, $w_1+w_2\in[S_1\cup S_2]$.