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??