Proving intersection is a subspace

Let and be two subspaces of a vector space over the field . Prove that the intersection of and is a subspace of .

Can someone check my working for this question

Let

Since is a subspace, it is closed under addition and so

Similarly, let

Since is a subspace, it is closed under addition and so

Closure under addition

Let

since is a subspace

since is a subspace

Clearly, Closure under scalar multiplication

is a subspace of

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?