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A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers .
Then has an upper bound (, for example, or ) but no least upper bound in : If we suppose is the least upper bound, a contradiction is immediately deduced because between any two reals and (including and ) there exists some rational , which itself would have to be the least upper bound (if >) or a member of greater than (if <).
By the least-upper-bound property of real numbers, = {} exists and . Now, for every ε > 0 {\displaystyle \varepsilon >0} , there exists N {\displaystyle N} such that c ≥ a N > c − ε {\displaystyle c\geq a_{N}>c-\varepsilon } , since otherwise c − ε {\displaystyle c-\varepsilon } is a strictly smaller upper bound of { a n ...
In fact, if this were false, then the integers would have a least upper bound N; then, N – 1 would not be an upper bound, and there would be an integer n such that n > N – 1, and thus n + 1 > N, which is a contradiction with the upper-bound property of N. The real numbers are uniquely specified by the above properties.
The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers = {<}. This set has an upper bound. However, this set has no least upper bound in Q: the least upper bound as a subset of the reals would be √2, but it does not exist in Q.
Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: the smallest number that is greater than all of the terms of the sequence. One of the axioms of the real number system is the completeness axiom, which states that every bounded sequence has a least upper bound.
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Chebyshev's inequality then follows by dividing by k 2 σ 2. This proof also shows why the bounds are quite loose in typical cases: the conditional expectation on the event where |X − μ| < kσ is thrown away, and the lower bound of k 2 σ 2 on the event |X − μ| ≥ kσ can be quite poor.