Is V Perpendicular a Subset of U Perpendicular in Subspaces of Rn?

[stunner5000pt]

Suppose U, V are proper subsets of Rn and are subspaces and U is a proper subset of V. PRove that V perp is a proper subset of U perp

Ok SO let U ={u1, u2, ..., un}
let V = {v1,...vn}
let V perp = {x1,x2,..., xn}
let U perp = {w1,...wn}
certainly u1 . w1 = 0
(u1 + u2 ) . (w1+w2) = 0
cu1 . cw1 = 0

everything till now is pretty much basically understood
i can't find a wayu to maek a proof work... Any hints would be appreciated!
I really want to be able to get this one!

 

[stunner5000pt]

can anyone help?

i am still stuck on this problem and i need to solve it ...

ok so far i get this part
if U perp is orthognal to U then U perp is orthogonal to V.
If V perp is orthognal to V then V perp is orthogonal to U
this V perp is orthogonal to U perp

then Wn . Xn = 0
now U^{\bot} \bigcap U = \{0\}
and U^{\bot} \bigcap V = \{0\}
similarly it applies for V perp
so both U perp and V perp include elements that are perpendicular to U and V and contain everything (but zero) that is perpendicular to U and V, thus not included in U and V.
Now how would i relate U perp and V perp to each other. Is there a theorem that says that the perpendicular space to a bigger subspace is smaller than the perpendicular subspace to the subspace contained in the bigger subspace??