The Archimedean property says that the natural numbers are unbounded in the real numbers. In particular, for every positive real number there is a positive integer with . Otherwise a purported upper bound for all natural numbers contradicts the least-upper-bound property by translating that bound by one.
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The Archimedean property is a fundamental concept in mathematics that relates to the behavior of real numbers, particularly in the context of the ordering of numbers. It states that for any two positive real numbers \( a \) and \( b \), there exists a natural number \( n \) such that: \[ n \cdot a > b.