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Similarly, a function g defined on domain D and having the same codomain (K, ≤) is an upper bound of f, if g(x) ≥ f (x) for each x in D. The function g is further said to be an upper bound of a set of functions, if it is an upper bound of each function in that set.
In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral.If f is a complex-valued, continuous function on the contour Γ and if its absolute value | f (z) | is bounded by a constant M for all z on Γ, then
By the boundedness theorem, f is bounded from above, hence, by the Dedekind-completeness of the real numbers, the least upper bound (supremum) M of f exists. It is necessary to find a point d in [a, b] such that M = f(d). Let n be a natural number. As M is the least upper bound, M – 1/n is not an upper bound for f.
of a Riemann integrable function defined on a closed and bounded interval are the real numbers and , in which is called the lower limit and the upper limit. The region that is bounded can be seen as the area inside a {\displaystyle a} and b {\displaystyle b} .
Similarly, the set of integers has the least-upper-bound property; if is a nonempty subset of and there is some number such that every element of is less than or equal to , then there is a least upper bound for , an integer that is an upper bound for and is less than or equal to every other upper bound for .
The Q-function is not an elementary function. However, it can be upper and lower bounded as, [6] [7] ... , the best upper bound is given by = and = with maximum ...
By the least-upper-bound property, S has a least upper bound c ∈ [a, b]. Hence, c is itself an element of some open set U α, and it follows for c < b that [a, c + δ] can be covered by finitely many U α for some sufficiently small δ > 0. This proves that c + δ ∈ S and c is not an upper bound for S. Consequently, c = b.
Applied to Cauchy's bound, this gives the upper bound + = for the real roots of a polynomial with real coefficients. If this bound is not greater than 1, this means that all nonzero coefficients have the same sign, and that there is no positive root.