📂Analysis

Principle of Archimedes in Analysis

Theorem

For any positive number $a$ and real number $b$, there exists a natural number $n$ that satisfies $an>b$.

Explanation

This means that no matter what $b$ you take, you can always think of a multiple of $a$, which is $n$, that is greater than it. Simply put, ‘No matter how small a number is, if you keep adding to it, it will continue to grow’ is a very common-sense and obvious principle.

It has nothing to do with the principle of buoyancy or Eureka; it just shares the name.

Proof

Strategy: The proof process mobilizes the three axioms of analysis. Even something that seems so obvious is meticulously completed by precisely mentioning those axioms.

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• Case 1

  If $a>b$, then $an>b$ is satisfied when $n=1$.

• Case 2

  Let’s say $E := \left\{ n \in {\mathbb{N}} ,|, an<b \right\}$. By the axiom of inverses, the inverse $\dfrac{1}{a}$ of $a$ exists, and by the axiom of order, $\dfrac{1}{a}>0$. Therefore, the following holds true:

  $$an<b \iff n < \dfrac{b}{a}$$

  That is, $E = \left\{ n \in {\mathbb{N}} ,|, n < \dfrac{b}{a} \right\}$ is bounded above. By the axiom of completeness, since $\sup(E)$ exists, $n=\sup(E)+1$ exists that satisfies $an>b$.

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