The two convergent subsequences of the sequence (-1)" +2 is
a) x = (-1)" +2 and 2,4 =[(-1)] +2
b) x2n = (-1)"" + 2 and x2n-1
c) Xzn = (-1)"" +2 and xxn1 = (-1) 29+1 +2
d) x2 = (-1)" +2 and Xzov = (-1)" +2
2n+1
+2
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Answer to Question #175361 in Real Analysis for Anand
Answers>Math>Real Analysis
Question #175361
Give an example of a divergent sequence which has two convergent sequences. Justify your claim.
Expert's answer
Consider the sequence (−1)^n(−1)
n
which is non convergent or a divergent sequence.
You can argue that it has a convergent subsequence in two ways:
(1) It is bounded, so by the Bolzano Weirstrass Theorem, it has a convergent subsequence.
(2) Observe that the subsequence, (1,1,1,1,1,1,1...) is convergent (since it is constant).