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'''Relation to limits of sequences'''
If <math>S \neq \varnothing</math> is any non-empty set of real numbers then there always exists a non-decreasing sequence <math>s_\bull = \left(s_n\right)_{n=1}^\infty</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \sup S.</math> Similarly, there will exist a (possibly different) non-increasing sequence <math>s_\bull</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \inf S.</math>
Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact from [[topology]] that if <math>f</math> is a [[Continuous function (topology)|continuous function]] and <math>s_1, s_2, \ldots</math> is a sequence of points in its domain that converges to a point <math>p,</math> then <math>f\left(s_1\right), f\left(s_2\right), \ldots</math> necessarily converges to <math>f(p).</math>
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