# Problems with some exercices

1. Sep 14, 2011

### 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.

Last edited: Sep 14, 2011
2. Sep 14, 2011

### micromass

Try to use that

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

and hence

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

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??