Span(S1 ∩ S2) ⊆ span(S1) ∩ span(S2)

[Luisito89]

Hi! I wonder if you guys help me prove this:
span(S1 ∩ S2) ⊆ span(S1) ∩ span(S2)
I have seen span(S1 U S2) = span(S1) + span(S2), so I just want to prove this one, any suggestion?

 

[MathematicalPhysicist]

v=as, s in S1 and S2, thus v is in sp(S1) and sp(S2).
Vice versa isn't necessarily because if v in sp(S1) and sp(S2), then v=as1=bs2, and s1 isn't necessarily in S2 and s2 isn't necessarily in S1.