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The lower extreme is the lowest number in a set of data

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That is the correct conventional definition. However, for the mathematical definition you may have to allow for the fact that the lower extreme may not actually be attained: it may be an asymptotic value. For example, consider the set of numbers 1, 1/2, 1/4, 1/8 and so on. The sequence gets closer to 0 as you keep going but it is never actually equal to 0. The lower extreme of this infinite set is 0, but that is not an element of the set.

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Q: What is the math definition for lower extreme?
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