Non-Empty Bounded Above Subset of Banach Space with Archimedean Property has Supremum

Theorem

Let $\BB$ be a Banach space.

Let $\BB$ have the Archimedean property.

Let $S \subseteq \BB$ be a subset of $\BB$ which is bounded above.

Then $S$ admits 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 $1.2.10$