Today I Learned

Some of the things I've learned every day since Oct 10, 2016

97: The Archimedean Property of the Reals

\mathbb{R} has the Archimedean property, which states that for any positive x, y \in \mathbb{R} there exists an n \in \mathbb{N} such that

nx > y.

Intuitively, this means that \mathbb{R} contains neither infinitely large nor infinitesimally small numbers. (The property would not hold if y was infinite, or if x was infinitesimal.)

Proof:

Finding an n such that nx > y is equivalent to finding one such that n > \frac{y}{x}. If there is no such n, this means that n \leq \frac{y}{x} for all n and thus the number \frac{y}{x} is an upper bound on \mathbb{N}. However, \mathbb{N} has no upper bound, so this is a contradiction, meaning such an n must exist.

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