Theorem – Principle of Archimedes

Contents
1. Theorem
2. Corollaries
3. Extension to a series of real numbers

1. Theorem

The set $\mathbb{N}$ is unbounded above in $\mathbb{R}$ .

Proof –

We have that if $\mathbb{N}$ were bounded above in $\mathbb{R}$ , then there would exist some upper bound for $\mathbb{N}$ , say $r\in\mathbb{R}$ such that for all $n\in\mathbb{N}$ ,

$n < r$

Now let $S$ be the set of upper bounds for $\mathbb{N}$ . Then, let $r' = \text{inf}(S)$ . If $r'$ is a natural number, then let $r'' = r'+1$ . Thus, it follows that $r''>r'$ , and as both $r''$ and $r'$ are natural numbers, and since $r''\in\mathbb{N}$ , $r'$ cannot be an upper bound for $\mathbb{N}$ in $\mathbb{R}$ .

Now, if $r' \in \mathbb{R}^+ \setminus \mathbb{N}$ , then simply take the upper bound, $M$ , to be $r''$ rounded down to the nearest natural number. Then, we have that $r''+1$ is larger than $M$ , but is also an element of the natural numbers, and so $M$ cannot be an upper bound for $\mathbb{N}$ in $\mathbb{R}\quad\square$

2. Corollaries

There are several corollaries which can be obtained via the above theorem,

For every $r\in\mathbb{R}$ , there exists some $n\in\mathbb{N}$ such that $r<n$ .
Let $x >0$ be a real number and let $y\in\mathbb{R}$ . There exists some $n\in\mathbb{N}$ such that $nx > y$ .
For each $x>0$ , there exists $n\in\mathbb{N}$ such that $o<\frac{1}{n}<x$ .