Proof: Intersection of Subspaces H and K is a Subspace of Vector Space V

[Bds_Css]

I am having trouble with this proof:

Let H and K be subspaces of the vector space V. The intersection of H and K, written as
H \cap K, the set of v in V that belong to both H and K. Show that
H\cap K is a subspace of V

 

[ZioX]

What part are you having trouble with? Remember, if H and K are subspaces of the vector space V you must show that:

1) 0 is in HnK
2) For x,y in Hnk, x+y in HnK
3) k a scalar (in the relevant field) and x in HnK, kx in HnK