Theorem

If V is finite-dimensional and U is a subspace of V, then dim U <= dim V.

Proof

Suppose V is finite-dimensional and U is a subspace of V. Think of a basis of U as a linearly independent list in V, and think of a basis of V as a spanning list in V. Thus, we can use 1753232272 - Axler 2.22 Length of linearly independent list <= length of spanning list|2.22 to show that dim U <= dim V.