Show that for any subset $S \subset V$ that the span(S) is the smallest subspace of $V$ containing $S$.

The fact that $\text{span}(S)$ is a subspace of $V$ is OK. ($0$ being an element need not be verified, it follows by taking the scalar $0$.) Remark that it clearly contains $S$ as well (for any $s \in S$ take the trivial $1\cdot s \in \text{span}(S)$.

The proof that it is the smallest such: suppose that $W$ is any linear subspace of $V$ that contains $S$. Then any member $x \in \text{span}(S)$, i.e. any $x=\alpha_1 s_1 + \ldots \alpha_n s_n$, for some finitely many $s_i \in S$ and scalars $\alpha_i$, is in $W$ as all $s_i$ are ($S \subseteq W$ by assumption, and $W$ is closed under scalar multiplication and addition, being a subspace), so $\text{span}(S) \subseteq W$. This holds for any such $S$ and that is the definition of being the smallest subspace of $V$ that contains $S$.

Your question is about proving that two sets are equal. For such problem, I would suggest to name the two sets. In your case, let's say $S_1 = \text{span} (S)$ and $S_2$ the smallest linear subspace containing $S$.

Then to prove that two sets are equal, a usual way is to prove that $S_1 \subseteq S_2$ and $S_2 \subseteq S_1$ by presenting the proofs for both inclusion.

You proved in the first part that $\text{span}(S)$ is a subspace and this indeed implies that $S_2 \subseteq S_1$.

Now you have to prove that $S_1 \subseteq S_2$.

To prove that it is sufficient to proof that any subspace containing $S$ contains $\text{span}(S)$. Take a subspace $S \subseteq S^\prime$ and $s \in \text{span}(S)$. By definition, it exists $s_1, \dots, s_n \in S$ and $\lambda_1, \dots, \lambda_n \in \mathbb K$ such that $s=\lambda_1 s_1+ \dots \lambda_n s_n$. And as $S^\prime$ is a subspace, it contains $s$. Allowing to reach the desired conclusion.