Proof: Archimedean Principle of Real Numbers | Real Analysis

Given real numbers a and b, where a is positive, we can always find a natural number m so that n*a is greater than b. In other words, we can add a to itself enough times to get a number greater than b. Equivalently, given any real number x, there exists a natural number greater than x, meaning the natural numbers are unbounded above. This is the Archimedean Principle, and we prove it in today's real analysis lesson. #realanalysis

This is a proof by contradiction, making use of the definition of supremum and the completeness axiom/least upper bound property.

Definition of Supremum and Infimum:    • Definition of Supremum and Infimum of...  

Real Analysis Course:    • Real Analysis  
Real Analysis exercises:    • Real Analysis Exercises  

Other similar names for this principle: Archimedean Property, Archimede's Principle, Archimedean Axiom, axiom of Archimedes

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