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Problems with some exercices

  1. Sep 14, 2011 #1
    Hi everyone, I've had some troubles to solve some exercices of real analysis.


    1. Prove that [tex] card( \mathbb{R}^{\mathbb{N}}) = card(\mathbb{R})[/tex].

    In this one I have considered that [tex] card(0,1)= card(\mathbb{R}) [/tex] and tried to construct a bijection [tex] f: (0,1)\rightarrow \mathbb{R}^{\mathbb{N}}[/tex].


    2. Construct a bijection between [tex] [0,1]^{2} [/tex] and [tex] \mathbb{R} [/tex]

    -Thanks.
     
    Last edited: Sep 14, 2011
  2. jcsd
  3. Sep 14, 2011 #2

    micromass

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    Try to use that

    [tex]\mathbb{R}=2^\mathbb{N}[/tex]

    and hence

    [tex]\mathbb{R}^\mathbb{N}=2^{\mathbb{N}\times \mathbb{N}}[/tex]

    It might be easy to first construct a bijection between [itex][0,1]^2[/itex] and [0,1]. Try to do something with the decimal representation here. Given

    [tex]0.x_1x_2x_3...~\text{and}~0.y_1y_2y_3...[/tex]

    how could you combine these two numbers??
     
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