Prove that for every $ \quad n\in\mathbb{N},\quad \mathbb{R}^{n} = \mathfrak{c}$?

Question

I was thinking about induction like: 

Base:
$$\#\mathbb{R}^{1} = \#\mathbb{R} = \mathfrak{c}$$
And for $n+1$
$$\#\mathbb{R}^{n+1} = \#\mathbb{R}^{n}\mathbb{R} = \mathfrak{c}$$
But it seems too easy. Do you have some other ideas?

Answer 1

If you already know that $\Bbb{R\times R}$ and $\Bbb R$ have the same cardinality, then this is enough.