## Cardinality of Reals

Consider the cardinality of the real numbers. Here we will prove the real numbers are an infinite set that are uncountable. Let $f(x)=\frac{1}{1+n}| n \in \mathbb{N}.$ Here you can see that all the fractions in this sequence are between 0 and 1. It is also apparent that each fraction corresponds to a natural number. In that sense each natural number has been paired with a fraction. There are as many fractions as there are natural numbers. Yet this set of fractions is only a subset of the real numbers, therefore, the real numbers are not countably infinite. The real numbers are uncountable. We say the real numbers have a cardinal number $c$. You can also see that the real numbers on the open interval (0,1) has a cardinal number greater than $\aleph_0$ because each fraction is between 0 and 1.

It can also be proven any subset of $\mathbb{R}$ with an open interval has a cardinal number $c$. Suppose (a,b) is an open interval. We can create a function that will map the open interval (0,1) onto (a,b) which means that the interval (a,b) has a cardinal number $c$. $f(x)=a + (b-a)x$ is just that function. We can see the left endpoint has been mapped to a and the right endpoint has been mapped to b.