Exercise · ZW-SUPREMUM-AND-INFIMUM-009 · ★★★ Advanced
Let \(M\) be an upper bound of \(S\). When is \(M=\sup S\)?
Choose one option.
Show answer
- C. When for every \(\varepsilon>0\) there is \(s\in S\) with \(s>M-\varepsilon\)
Explanation
No smaller number may be an upper bound: there are always elements of \(S\) arbitrarily close to \(M\) from below.
Review the concept: Supremum and infimum
Given a set \(S\) of real numbers, its supremum is the least of its upper bounds and its infimum is the greatest of its lower bounds.
\[s\le\sup S,\quad s\ge\inf S\qquad\forall s\in S\]
More practice problems: supremum and infimum
- ZW-SUPREMUM-AND-INFIMUM-001 What is the supremum of the interval (0,1)?
- ZW-SUPREMUM-AND-INFIMUM-002 What is the infimum of [2,7)?
- ZW-SUPREMUM-AND-INFIMUM-003 What is the maximum of (0,1)?
- ZW-SUPREMUM-AND-INFIMUM-004 Find sup\1-1/n : n∈mathbb N\ with n≥1.
- ZW-SUPREMUM-AND-INFIMUM-005 Find inf\1/n : n∈mathbb N\ with n≥1.
- ZW-SUPREMUM-AND-INFIMUM-006 Find sup\x∈mathbb R : x^2<2\.
- ZW-SUPREMUM-AND-INFIMUM-007 What is the supremum of (0,1)∪[2,3)?
- ZW-SUPREMUM-AND-INFIMUM-008 If S=(1,4) and -S=\-s : s∈ S\, what is sup(-S)?