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
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
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 .