Examples
2 and 5 are both lower bounds for the set { 5, 10, 34, 13934 }, but 8 is not. 42 is both an upper and a lower bound for the set { 42 }; all other numbers are either an upper bound or a lower bound for that set.
Every subset of the natural numbers has a lower bound, since the natural numbers have a least element (0, or 1 depending on the exact definition of natural numbers). An infinite subset of the natural numbers cannot be bounded from above. An infinite subset of the integers may be bounded from below or bounded from above, but not both. An infinite subset of the rational numbers may or may not be bounded from below and may or may not be bounded from above.
Every finite subset of a totally ordered set has both upper and lower bounds.
Read more about this topic: Upper And Lower Bounds
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