Archimedes Principle

This is a statement that for any real number $x$ there is an integer $n$ larger than $x$ .

Assume no such $n$ exists. Then $x$ is an upper bound for the set of integers, which is a nonempty subset of the real numbers. Hence this set has a least upper bound $y$ that is a real number. But then there is an $n$ such that $y - 1/2 \le n$ . and so $y < y+1/2 \le n+1$ . But since $n+1 \le y$ , it follows that a contradiction is reached and therefore the assumption is false.

Using AP to Establish Density of Rationals in the Reals

Let $x,y$ be positive real numbers such that $x < y$ . Choose integer $n$ such that $(y - x)n > 1$ , which exists by AP. Let $M$ be the set of all natural numbers $m$ for which $m \ge ny$ . By AP, this set is nonempty, and by WOP this set contains a smallest element $m$ . Hence $(m-1) < ny \le m$ . But since $yn - 1> xn$ , it follows that $xn < m - 1$ and so $x < \frac{m-1}{n} < y$ .

Archimedes Principle

Using AP to Establish Density of Rationals in the Reals

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