Let $\mathbb{R}^{n}$ be the affine $n$-space. Let $\{V_{\alpha} \}_{\alpha \in \Gamma}$ be a set of real varieteis such that $\mathbb{R}^{n} \subseteq {\bigcup V_{\alpha}}.$ Prove that $|\Gamma| \geq 2^{\aleph_{0}}$
I know that one can take the set $A =\{(t^{a_{1}},...,t^{a_{n}}) | t \in [0,1],\hspace{0.2cm} a_{1},...,a_{n} \text{ such that they are linearly independent over} \hspace{0.2cm}\mathbb{Q}\}$ and show that every polynomial intersects with this set in finitly many points and prove the statement.
But now does anyone know rather a different proof of this statement?