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Prove that $|\mathbb{R}^n| = |\mathbb{R}|$.

It will be enough to prove $|\mathbb{R}^{2}|=|\mathbb{R}|$. We can further simplify by proving $|(0,1)\times(0,1)| = |(0,1)|$ (because $|\mathbb{R}|=|(0,1)|$).

BUT how to proceed further?

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  • $\begingroup$ Do you mean $\Bbb R$ to be where you put IR? $\endgroup$ – Wojowu Aug 16 '15 at 11:14
  • $\begingroup$ Schröder-Bernstein-Cantor will help you here, but then you need to find an injection from $\mathbb{R}^2\to\mathbb{R}$ $\endgroup$ – user190080 Aug 16 '15 at 11:14
  • $\begingroup$ Yes it means R set of real numbers $\endgroup$ – Vishavjeet Singh Aug 16 '15 at 14:46