nested intervals

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nested intervals

[′nes·təd ′in·tər·vəlz]
(mathematics)
A sequence of intervals, each of which is contained in the preceding interval.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Since for any [lambda] [member of] A, [mathematical expression not reproducible] is a closed interval [mathematical expression not reproducible], by the definition of subset of soft sets [mathematical expression not reproducible] is a sequence of sets satisfying the properties of nested intervals theorem. Hence [[??].sup.[infinity].sub.n=1] [I.sub.n]([lambda]) = {[a.sub.[lambda]]}, which implies [mathematical expression not reproducible] (where [??]([lambda]) = [a.sub.[lambda]], [for all][lambda] [member of] A).
Clearly the sequence {[I.sub.n]}, where [mathematical expression not reproducible] for all n [member of] N, satisfies the nested intervals theorem. Therefore, there exists a unique [mathematical expression not reproducible] and [[??].sub.n](v) [right arrow] z, [p.sub.n](v) [right arrow] z as n [right arrow] [infinity].
5.1 From the Vertical Line Segment to the Nested Intervals Theorem
After careful analysis, we can find here we use the mathematical method of dynamic approximation, more importantly, which contains nested intervals theorem in mathematical analysis.