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how to show that sets $\{0,1\}^{\mathbb{N}}$ and $\mathbb{R}^{\mathbb{N}}$ have an equal cardinality?

I tried to use Cantor-Bernstein theorem but it seems to be hard.

Asaf Karagila
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JohnCyna
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  • Hint: Do it in two steps: first show that ${0,1}^{\mathbb{N}}$ and $\mathbb{R}$ have the same cardinality, then that $\mathbb{R}$ and $\mathbb{R}^{\mathbb{N}}$ also share the same cardinality. Conclude by transitivity. – b00n heT Jan 07 '17 at 11:04
  • Hi, i'm trying to show that $\mathbb{R}$ have the same cardinality as $\mathbb{R}^{\mathbb{N}}$ – JohnCyna Jan 07 '17 at 11:10
  • @JohnCyna Note that ${0,1}^{\Bbb N}$ and ${0,1}^{\Bbb N\times \Bbb N}$ have same cardinality because $\Bbb N$ and $\Bbb N\times \Bbb R$ have – Hagen von Eitzen Jan 07 '17 at 12:07

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You may replace $\Bbb R$ with $[0,1)$. The elements of $[0,1)$ are represented as infinite binary sequences. You can get from sequences of sequences of binary digits to sequences of binary digits by a zig-zag-enumaration as in the diagonal argument.

Hagen von Eitzen
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