Some definitions worth keeping in mind while studying for the next of proofs.

Axiom of Completeness

Every nonempty set of real numbers that is bounded above has a least upper bound.


Upper and Lower Bounds

A set \(A \in \mathbf{R}\) is bounded above if there exists a number \(b \in \mathbf{R}\) such that \(a \leq b\) for \(a \in A\). The number \(b\) is an upper bound for \(A\). Similarly, the set \(A\) is bounded below if there exists a lower bound \(l \in \mathbf{R}\) satisfying \(l \leq a\) for every \(a \in A\).


Note here that \(b\) doesn’t need to be \(A\). A set like \(\{1,2\}\) can have an upper bound like 3 for example. Every element in this set is less than 3. Also note here that \(A\) and \(b\) are both in \(\mathbf{R}\).

Questions: What if we don’t find such a \(b\)? can we conclude that the set is infinite?

Least Upper and Greater Lower Bounds

A real number \(s\) is the least upper bound for a set \(A \in \mathbf{R}\) if it meets the following two criteria:
(i) \(s\) is an upper bound for A;
(ii) if \(b\) is any upper bound for \(A\), then \(s \leq b\).


The least upper bound is also called the supremum of the set \(A\) or \(\sup A\) while the greatest lower bound is called the infimum or \(\inf A\). Both bounds are unique. There can only be least upper bound and one greatest lower bound.

Questions: since \(s\) must be an upper bound, does it mean that the set \(A\) is a bounded set?

Minimum and Maximum

A real number \(a_0\) is a maximum of the set \(A\) if \(a_0\) is an element of \(A\) and \(a_0 \geq a\) for all \(a \in A\). Similarly a real number \(a_1\) is a minimum \(a_1\) is a minimum of the set \(A\) if \(a_1 \in A\) and \(a_1 \leq a\) for all \(a \in A\).


It’s definitely clear here that the maximum and minimum must be elements of the set itself unlike the upper and lower bounds.

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