Ordered Field with Archimedean Property on which Monotone Convergence Theorem Holds has Continuum Property

Theorem

Let $\FF$ be an ordered field on which the Archimedean property holds.

Let $\FF$ be such that every sequence which is increasing and bounded above is convergent.

Then every non-empty subset of $\FF$ which is bounded above has a supremum.

Proof

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Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Theorem $1.2.7$