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?
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.