Consider how 'dimension' interacts with 'subset'. Assume U and W are both subspaces of some vector space,

Question:

Consider how 'dimension' interacts with 'subset'. Assume U and W are both subspaces of some vector space, and that U ⊂ W.
(a) Prove that dim(U) 6 dim(W).
(b) Prove that equality of dimension holds if and only if U = W.
(c) Show that the prior item does not hold if they are infinite-dimensional.