A Subset of a Vector Space is a Subspace iff it contains all its Linear Combinations
# Statement
$W$ is a Vector Subspace of $V$ If and Only If $\forall n \in \mathbb{Z}{\geq 0}$, $x{1}, \dots x_{n} \in W$, $c_{1}, \dots, c_{n} \in F$, $c_{1} x_{1} + \cdots + c_{n} x_{n} \in W$.
# Proof
$(\Rightarrow)$ This follows from the definiton of a Vector Subspace, all the necessary operations stay within the Vector Subspace. $\checkmark$
($\Leftarrow$) Let $n = 2$ and choose $x_{1}, x_{2} \in W$ and $c \in F$. Then, $c x_{1} + 1 * x_{2}$ is a valid Linear Combination and is thus in $W$. Since A Subset of a Vector Space is a Subspace iff it is closed under scaling and addition, we have that $W$ is a Vector Subspace. $\checkmark$ $\blacksquare$