# 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 $$A$$.

Date: 2022-04-10 Sun 00:00