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This is a problem from ISI MStat 2018 PSA Problem 11 based on Sequence and subsequence.

Let be a sequence such that

Suppose the subsequence is bounded. Then

- (A) is always convergent but need not be convergent.
- (B) both and are always convergent and have the same limit.
- (C) is not necessarily convergent.
- (D) both and are always convergent but may have different limits.

Sequence

Subsequence

Answer: is (B)

ISI MStat 2018 PSA Problem 11

Introduction to Real Analysis by Bertle Sherbert

Given that is bounded . Again we have which shows that if is bounded then is also bounded .

Again both and both are monotonic sequence . Hence both converges .

Now we have to see whether they converges to same limit or not ?

As both and are bounded hence is bounded and it's already given that it is monotonic . Hence converges . So, it's subsequences must converges to same limit . Hence option (B) is correct .

This is a problem from ISI MStat 2018 PSA Problem 11 based on Sequence and subsequence.

Let be a sequence such that

Suppose the subsequence is bounded. Then

- (A) is always convergent but need not be convergent.
- (B) both and are always convergent and have the same limit.
- (C) is not necessarily convergent.
- (D) both and are always convergent but may have different limits.

Sequence

Subsequence

Answer: is (B)

ISI MStat 2018 PSA Problem 11

Introduction to Real Analysis by Bertle Sherbert

Given that is bounded . Again we have which shows that if is bounded then is also bounded .

Again both and both are monotonic sequence . Hence both converges .

Now we have to see whether they converges to same limit or not ?

As both and are bounded hence is bounded and it's already given that it is monotonic . Hence converges . So, it's subsequences must converges to same limit . Hence option (B) is correct .

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