1. ## Cardinality

Can anyone tell me how I could show that the following set is countable?

Natural numbers and the real numbers with decimal representations consisting of all 1's?

2. Originally Posted by vexiked
Natural numbers and the real numbers with decimal representations consisting of all 1's?
Have you copied the question exactly?

3. The question is as follows:

Prove that the sets of natural numbers and the real numbers with decimal representations of all 1's are countable or uncountable.

4. Originally Posted by vexiked
The question is as follows:

Prove that the sets of natural numbers and the real numbers with decimal representations of all 1's are countable or uncountable.
Do you mean the set containing the numbers like 1111.1111111?

Here's a hint: You can identify such a number with the couple $\displaystyle (m,n)\in\mathbb{N}^2$ where $\displaystyle m$ is the number of 1's on the left, and $\displaystyle n$ the number of 1's on the right. (Actually, you should probably also consider $\displaystyle n=\infty$ as a possibility.) From there, it should be clear whether the set is countable or uncountable.

5. Originally Posted by vexiked
The question is as follows:

Prove that the sets of natural numbers and the real numbers with decimal representations of all 1's are countable or uncountable.
$\displaystyle {\color{blue} \mathbb{N}}$ is countable by definition.