Proving intersection is a subspace

Let $\displaystyle W_1$ and $\displaystyle W_2$ be two subspaces of a vector space $\displaystyle V$ over the field $\displaystyle \mathbb{F}$. Prove that the intersection of $\displaystyle W_1$ and $\displaystyle W_2$ is a subspace of $\displaystyle V$.

Can someone check my working for this question

Let $\displaystyle x\in W_1, y\in W_1$

Since $\displaystyle W_1$ is a subspace, it is closed under addition and so $\displaystyle x+y\in W_1$

Similarly, let $\displaystyle x\in W_2, y\in W_2$

Since $\displaystyle W_2$ is a subspace, it is closed under addition and so $\displaystyle x+y\in W_2$

$\displaystyle \Rightarrow x+y\in W_1\cap W_2\Rightarrow$ Closure under addition

Let $\displaystyle \lambda\in\mathbb{R}, x\in W_1, W_2$

$\displaystyle \lambda x\in W_1$ since $\displaystyle W_1$ is a subspace

$\displaystyle \lambda x\in W_2$ since $\displaystyle W_2$ is a subspace

Clearly, $\displaystyle \lambda x\in W_1\cap W_2\Rightarrow$ Closure under scalar multiplication

$\displaystyle \Therefore W_1 \cap W_2$ is a subspace of $\displaystyle V$

What about the condition for non-empty? This condition always confuses me. Why is it that the zero vector has to be included for it to be non-empty?