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Sequences, limits and continuity
Published in Alan Jeffrey, Mathematics, 2004
Since we have not insisted that there be a finite number of points outside any neighbourhood of a limit point it follows that a sequence may have more than one limit point. We shall show by example that a limit point may or may not be a member of the sequence that defines it. This result, when applied to sequences with only one limit point, will later be seen to be very important, since it provides the justification for the approximation to irrational numbers in calculations by rational numbers. In sequences involving only one limit point the sequence will be said to converge to the value associated with the limit point. This value will be called the limit of the sequence.
Calculus Problems
Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018
SolutionThe procedures of finding the limit of a sequence are exactly the same as the limit of a function. The symbolic variables must be declared first, then, the sequence should be expressed as a symbolic expression. Finally the functionmust be called to get the results. For this example, the following commands should be issued, and the result is 0.
Painlevé–Kuratowski convergences of the solution sets for set optimization problems with cone-quasiconnectedness
Published in Optimization, 2022
Next, let us recall the concept of the Painlev–Kuratowski set convergence (see, e.g. [19]). Let be a sequence of subsets of . Set The set is called the upper limit of the sequence , and the set is called the lower limit of the sequence . We say that the sequence converges in the sense of Painlev –Kuratowski to the set A if We denote the Painlev–Kuratowski convergence by .
Generalized suzuki-type mappings in modular vector spaces
Published in Optimization, 2019
Andreea Bejenaru, Mihai Postolache
Let ρ be a convex modular with -property such that satisfies the modular -Opial condition. Let T be a modular -Suzuki mapping on a nonempty subset C, for a given admissible set of parameters. If is the -limit of a sequence and , then .
Confidence regions of stochastic variational inequalities: error bound approach
Published in Optimization, 2022
(i). Denote as the limit point of sequence . By the assumption that converges to on uniformly, as and Then is a solution to SVIP (3) w.p.1.