Prove that $\mathbb R^n$ is not the union of finitely many its proper subspaces. I am reading An Introduction to Algebraic Topology by Rotman. After proving 
Theorem 2.7: For every $k\geq 0$, euclidean space$\mathbb R^n$ contains $k$ points in general position,
the book remarked: There are other proofs of this theorem using induction on k. The key geometric observation needed is that $\mathbb R^n$ is not the union of only finitely many proper affine subsets .
I want to prove that observation.

Every affine proper subspace of $\mathbb{R}^n$ i contained in an affine hyperplane. 
Now, let $C=\{(t,t^2,t^3,\ldots,t^n),\,t \in \mathbb{R}^n\}$. 
Since every nonzero polynomial with degree at most $n$ has at most $n$ roots, an affine hyperplane of $\mathbb{R}^n$ contains at most $n$ points in $C$. 
So if you have a covering of $\mathbb{R}^n$ by proper affine subsets, you need at least $|\mathbb{R}|$ of them.
