Abstract:
is bounded from below if there is a number M such that s #S. The completeness axiom states that every nonempty subset of R that is bounded from below has a greatest lower bound. We will call it infimum of and denote it by inf . For example the closed and open intervals [a, and (a, b) have the same infimum, which is a.Now,ifS contains an element that is smaller than all its other elements, this element is called minimum of and is denoted by minS. Note that the minimum of a set does not always exist. For example, (a, b) has no minimum since a/# (a, b). On the other hand, if the minimum of a set exists, it is identical to its infimum. For example, min[a, b]1 nf[a, b] a. One easily shows that every finite nonempty subset of R has a minimum. Finally, let us mention that we will often use the notation to denote infimum (or, when it exists, the minimum). For example, a b =min{a, b}.IfS is empty, we adopt the convention that inf =+#. If f is a function from to R,
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