Let \(S\) be a set of real numbers. If there exists a real number \(b\) such that \(x\geq b\) for every \(x \in S\), then \(b\) is called a lower bound for \(S\) and we say that \(S\) is bounded below by \(b\).
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We say “a” lower bound because any number less than \(b\) will also be a lower bound.
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If \(b\) is also an element of \(S\) then it is called the minimum element of \(S\).
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A set with no lower bound is called unbounded below.
The infimum of the set \(S\) is the greatest of all the lowerbounds of the set. It is denoted as \(inf(S)\)