Let \(S\) be a set of real numbers. If there exists a real number \(b\) such that \(x\leq b\) for every \(x \in S\), then \(b\) is called an upper bound for \(S\) and we say that \(S\) is bounded above by \(b\).
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We say “an” upper bound because any number greater than \(b\) will also be an upper bound.
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If \(b\) is also an element of \(S\) then it is called the maximum element of \(S\).
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A set with no upper bound is called unbounded above.
The supremum of the set \(S\) is the least of all the upperbounds of the set. It is denoted as \(sup(S)\)