Constructing a bijection between [0,1]^2 and [0,1] using decimal representation.

[Hurin]

Hi everyone, I've had some troubles to solve some exercices of real analysis.1. Prove that card( \mathbb{R}^{\mathbb{N}}) = card(\mathbb{R}).

In this one I have considered that card(0,1)= card(\mathbb{R}) and tried to construct a bijection f: (0,1)\rightarrow \mathbb{R}^{\mathbb{N}}.2. Construct a bijection between [0,1]^{2} and \mathbb{R}

-Thanks.

 

[micromass]

Hi everyone, I've had some troubles to solve some exercices of real analysis.


1. Prove that card( \mathbb{R}^{\mathbb{N}}) = card(\mathbb{R}).

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

Try to use that

\mathbb{R}=2^\mathbb{N}

and hence

\mathbb{R}^\mathbb{N}=2^{\mathbb{N}\times \mathbb{N}}

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

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

0.x_1x_2x_3...~\text{and}~0.y_1y_2y_3...

how could you combine these two numbers??