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For example, 5 is a lower bound for the set S = {5, 8, 42, 34, 13934} (as a subset of the integers or of the real numbers, etc.), and so is 4. On the other hand, 6 is not a lower bound for S since it is not smaller than every element in S. 13934 and other numbers x such that x ≥ 13934 would be an upper bound for S.
Since it is an upper bound of C, it is the least upper bound of C. Dfeuer 22:44, 28 November 2007 (UTC) Thanks for your work. My primary concern is that the first part of the article be kept elementary, the way it is now. As far as everything else, there are now upper bounds. :) Oleg Alexandrov 04:19, 29 November 2007 (UTC)
Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.
Simpler lower and upper bounds for Frobenius numbers for n = 3 have also been determined. The asymptotic lower bound due to Davison The asymptotic lower bound due to Davison f ( a 1 , a 2 , a 3 ) ≡ g ( a 1 , a 2 , a 3 ) + a 1 + a 2 + a 3 ≥ 3 a 1 a 2 a 3 {\displaystyle f(a_{1},a_{2},a_{3})\equiv g(a_{1},a_{2},a_{3})+a_{1}+a_{2}+a_{3}\geq ...
Later calculations suggest that the cross section of the 3n reaction (which would result in 299 119 and three neutrons as products) would actually be six hundred thousand times lower than this upper bound, at 0.5 pb.
The construction follows a recursion by starting with any number , that is not an upper bound (e.g. =, where and an arbitrary upper bound of ). Given I n = [ a n , b n ] {\displaystyle I_{n}=[a_{n},b_{n}]} for some n ∈ N {\displaystyle n\in \mathbb {N} } one can compute the midpoint m n := a n + b n 2 {\displaystyle m_{n}:={\frac {a_{n}+b_{n ...
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