Suppose $U_1,\dots,U_m$ are finite dimensional subspaces of $V$. Prove that $U_1 + \dots + U_m$ is finite dimensional and

Let $u^j_1,\dots,u^j_{\dim U_j}$ be a basis of $U_j$ for all $1 \le j \le m$. Concatinating all the bases gives a list of length $\dim U_1 + \dots + \dim U_m$ which spans $U_1 + \dots + U_m$ thus it's finite dimensional. Then apply 2.31 to remove dependent vectors until we have a basis for $U_1 + \dots + U_m$. Thus