Supremum and Infimum Properties

Real Analysis · Axiom Academy

A bounded set may have no largest element — yet it always has a least upper bound . That single guarantee is what makes ℝ complete. Take a set . A number M is an upper bound if every element sits at or below it: for all . A set has infinitely many upper bounds — anything bigger than one is another. The supremum is the smallest one of them all. for every (it is an upper bound), and if M' is any upper bound of S , then (it is the least). The least-upper-bound condition is awkward to use directly — it quantifies over all other bounds. There is a cleaner, equivalent test: is the one upper bound you can get arbitrarily close to from inside S . an upper bound you can approach within any margin for every — nothing in the set exceeds it. For each there is an with , so is not an upper bound. To prove in practice you rarely compare against every bound. You show M bounds S , then for an arbitrary you produce a set element within of M . The infimum mirrors it: for each there is an with . 3. Completeness — the Defining Axiom of Does a least upper bound always exist? Not in . Let . It is bounded above (by 1.5 , say), but every rational upper bound can be nudged smaller — its edge sits exactly at , which is not rational . In the supremum is missing. The rationals with press toward , but no rational sits there. The least upper bound falls through the gap — it does not exist. The same set has , a genuine real number. Completeness guarantees the edge is always there to land on.

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