## 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...

## Cardinality

This page discusses cardinality, a basic topic found in classes like Sets, Relations and Functions and also Real Analysis. Definitions with discussion follow. numerically equivalent two sets are numerically equivalent if there exists a one-to-one function from one set onto the other. We can also say two sets have the same cardinal number if this is true. </dl> If we can say that a set has $n$ elements where $n$ is a specific element of the natural numbers, then we can say that particular set is countable. If each element in a infinite set can be paired with a natural...

## Cauchy Sequences

A sequence $a_n$ converges to $a$ if for every $\epsilon > 0$, $\exists$ an $N$ such that if $n > N$ then $\left| a_n - a \right| < \epsilon.$