Bounded Sets and Their Bounds

Real Analysis · Axiom Academy

A set on the number line can be fenced in from above by infinitely many values — but one of them is the tightest. Find it, and you've found the supremum. Every set has a tightest fence from above Real analysis is built on one quiet idea: a set of numbers can be hemmed in. Pin down the exact edge of where a set ends — even when the set never quite reaches that edge — and you have the single property, completeness, that separates the real line from the rationals. Watch the edge appear first, then go find it yourself. Here is the set S = [0, 5) shaded on the number line: every number from 0 up to (but not including) 5. Watch a candidate upper bound slide down from far above. Any value at or above 5 fences the set in — and as the candidate drops, the fence tightens, settling on the smallest value that still works. The fence can sit anywhere at or above 5 — but its tightest position, 5 itself, is the least upper bound. Test a value: is it an upper bound? A value u is an upper bound of S when no element of S climbs above it — that is, x u for every x S . Drag the marker. When it sits at or above the whole shaded set it locks green ; let any of the set slip past it and it flips red . Notice: once you're at or above 5, every value works. There are infinitely many upper bounds — so which one deserves a name? The least upper bound — the supremum

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