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Nilanjan Bhowmick AIR 3, CSIR NET (Earth Science)
Bhuvaneshwara posted an Question
December 15, 2019 • 16:49 pm
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2 points
2 points
- IIT JAM
- Mathematics (MA)
1 nas iniiilte no. OT imit poinis. Theorem Every limit point of the range set of a sequence is limit point of a sequence. Solution: Let S be range set of sequen
1 nas iniiilte no. OT imit poinis. Theorem Every limit point of the range set of a sequence is limit point of a sequence. Solution: Let S be range set of sequence {a.). i.e Let oa e S for e>0 range of fa, S (a E, a+ E) N S{a} has infinite no. of points Let q e (a - e, a + 6) N Sla qe (a- E, a+ e) and q e S qeS q a, for some k eN AS 01,0 1m So a, e (a - 8, a + 8) Tence, (a - 8, a + 6) Contains infinite no. of terms of sequenceorIt Proved a is limit point of sequence (a,) Remark A real no is a limit noint of sequence it appears in the sequence infinite many times
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Priya wadhwa![best-answer]()
It should be noted that every limit point l of the range set R{u} of a sequence u is also a limit point of the sequence u , because every neighborhood of l contains infinitely many points of R{u}, and so of the sequence u. On the other hand, a limit point of u may or may nor be a limit point of R{u}. If a sequence u is defined by u_n=1, then 1 is the only limit point of sequence.