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Nilanjan Bhowmick AIR 3, CSIR NET (Earth Science)
L.sufiya Khanam posted an Question
October 09, 2020 • 11:49 am
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30 points
30 points
- IIT JAM
- Mathematics (MA)
Plz explain me in an easy way what this theorem says and an example
utTEdSIn9, TEaSITg, TetX = Ini S and proce Ini Sand proceed in a sir TeL lar mann Lemma (On nested intervals). Let the sequence of the intervals {[a, b]} be such that [a b cla, b] Vn e N (such intervals are called nested), and their lengths converge to zero, i.e., lim n- (b- a)= 0. Then there exists a unique point that belongs to every interval, i.e., e la, b] vn e N, and 5- lim a, = sup{a,) = lim b, = inifb,}. no n0
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Srinath Best Answer
I hope you understand what nested intervals mean. Consider I1 = [a1,b1], a closed and bounded interval. Now, consider I2 = [a2, b2]. According to the conditions, the relation between I1 and I2 is that, I2 is contained in I1. That is, a1 <= a2 <= b2 <= b1. Similarly I3 is contained in I2, I4 in I3, and the pattern follows. I(n+1) is contained in I(n). Also, as n increases, with each In, the length of the intervals keep decreasing. Here's the heart of the theorem. Now it says that, the intersection of all such In, that is, intersection(I1 and I2 and....In...) contains only a single point. To get a rough visualisation of this, consider the example. Let I(n) = [ 0, 1/n]. Where n is in the set of all natural numbers. Check whether this interval satisfies the conditions of this theorem. Find an estimate as to what this "single point" would be, in the intersection of all I(n).
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Srinath
What did you not understand?
is this correct??
plz give me n another example
I understood what u explained but application??
What you wrote, is correct.
Application of Nested interval theorem? One thing that I know of, is that it is one of the fundamental properties that can be used to prove the completeness of R( real number system). Along with the Archimedean principle.
iam asking how to apply them in problems
You need to be more specific, because it depends on the problem, 😂
ok sir