Axiom of Archimedes
is not bounded above in : for every there exists with .
Proposition 5.6 (Axiom of Archimedes)
is not bounded above in .
Corollary
Corollary 5.7
For any real number , such that .
Proof
Suppose were bounded above and let . Then is not an upper bound, so some satisfies ; but then and , contradicting that is an upper bound. Consequently, for every real some exceeds , giving ; in particular , and there are no infinitely large or infinitely small real numbers.
Related
Stated in
- Proposition 5.6 (Axiom of Archimedes)§5.1 Construction of
- Corollary 5.7§5.1 Construction of