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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.
Euclidean and Hilbert spaces of functions
Published in Bertero Mario, Boccacci Patrizia, Introduction to Inverse Problems in Imaging, 2020
Bertero Mario, Boccacci Patrizia
Given a subset <S of X, a function / is a limit point of S if it is the limit of a sequence contained in S. A set S is said to be closed if it contains all its limit points. If S is not closed, it is possible to get a closed set by adding to S all limit points which do not belong to S. The new set is called the closure of S and denoted by S. It is the smallest closed set containing S.
Necessary conditions of extremum for functionals
Published in Simon Serovajsky, Optimization and Differentiation, 2017
The limit point of the sequence is the limit of some of its subsequence. If a sequence converges, then it has a unique limit point; it is equal to the limit of the sequence. If v is limit a point of all subsequences of the sequence, then this is the limit of the sequence.
Non-linear PI regulators in control problems for holonomic mechanical systems
Published in Systems Science & Control Engineering, 2018
Aleksandr Andreev, Olga Peregudova
The function () is monotonically decreasing along the solution due to the condition (10). Therefore, the following holds From the inequality (9) for all T>0 one can find that Let be a positive limit set and be a positive limit point defined by the sequence . As in the proof of Theorem 2.1 one can find the solution of the Equation (7) which passes through the point p, . Thereafter, for the sequences and constructed in Theorem 2.1 we have Hence, passing to the limit in (16) for and using (14) we obtain for all and correspondingly .
On the optimal relaxation parameters of Krasnosel'ski–Mann iteration
Published in Optimization, 2021
Songnian He, Qiao-Li Dong, Hanlin Tian, Xiao-Huan Li
Let H be a real Hilbert space with inner product and norm , respectively. We denote by I the identity mapping from H into itself and denote by and weak convergence and strong convergence of the sequence to the point x, respectively. Also, we denote by the weak limit point set of the sequence , i.e. The following identity is very easy to prove directly:
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.