Archimedes' Principle in Analysis
Theorem
For a positive number $a$ and a real number $b$, there exists a natural number $n$ satisfying $an>b$.
Explanation
It means that no matter what $b$ we take, we can always think of a multiple $n$ of $a$ that is larger than it. Simply put, it is a very commonsensical and obvious principle that ’no matter how small a number is, if you keep adding it, it keeps getting bigger'.
It has absolutely nothing to do with the principle of buoyancy or Eureka; only the name is the same.
Proof
Strategy: All three axioms of analysis are mobilized in the course of the proof. No matter how obvious the fact may seem, the key is to complete the proof meticulously while precisely citing those axioms.
Case 1
If $a>b$, then $an>b$ is satisfied when $n=1$.
Case 2
Let $E := \left\{ n \in {\mathbb{N}} ,|, an<b \right\}$. By the field axioms, the inverse $\dfrac{1}{a}$ of $a$ exists, and by the order axioms, $\dfrac{1}{a}>0$. Therefore, the following holds.
$$ an<b \iff n < \dfrac{b}{a} $$
That is, $E = \left\{ n \in {\mathbb{N}} ,|, n < \dfrac{b}{a} \right\}$ is bounded above. By the completeness axiom, $\sup(E)$ exists, so there exists $n=\sup(E)+1$ satisfying $an>b$.
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