Building a Subspace Complementary to Two Given Subspaces Lee's Introduction to Smooth Manifolds claims that given an n-dimensional vector space $V$ and two $k$-dimensional subspaces $P,P'$, we can find a single $n-k$-dimensional subspace $Q$ complementary to both $P$ and $P'$. How do we show this?

If $P=P'$ then there exists a $0 \neq v \in V \setminus P$, and then span$\{v\}$ is a subspace disjoint from $P$ and $P'$. If they are not equal, there exists a $v_1 \in P \setminus P'$ and a $v_2 \in P' \setminus P$. Then any non-trivial linear combination of these two will suffice to make a new subspace disjoint from $P$ and $P'$. I.e. span$\{v=av_1+bv_2\}$ for $a,b \neq 0$ will be a subspace (of dimension 1) disjoint from $P$ and $P'$. Now repeat this process with $P+v\Bbb R$ and $P' + v\Bbb R$, to get a new subspace of dimension $2$ and so on 
