NOTE: Archimedean Principle

The Archimedean principle is: If \(a\) and \(b\) are real numbers with \(a > 0\), then there exists a natural number \(n\) such that \(na > b\).

So \(n > \frac{b}{a}\), by this we can have a particular example.

Let \(a = \epsilon\) and \(b = 1\), then \(n > \frac{1}{\epsilon}\), and \(\epsilon > \frac{1}{n}\).

1. Example

Show that \(\inf \Big(\Big\{ \frac{1}{n} : n \in \mathbb{N} \Big\} \Big) = 0\).

1.1. Proof

Let \(A = \{\frac{1}{n} : n \in \mathbb{N}\}\). Since \(1\) and \(n\) are positive for each \(n \in \mathbb{N}\), shows \(\frac{1}{n} > 0\), so \(0\) is a lower bound of \(A\).

Let \(\epsilon > 0\), by Archimedean principle there exists some \(n \in \mathbb{N}\) such that \(\frac{1}{n} < \epsilon\). This element is in \(A\) and is less than \(0 + \epsilon\). Thus, \(0\) is infimum of \(A\) by definition: For all \(\epsilon > 0\), \(0 + \epsilon\) is not a lower bound of \(A\).