Question

If a sequence tn has limsuptn=t, there exists a subsequence converging to ___.

A. t

B. t+ϵ, where ϵ>0

C. Any value greater than t.

D. Any value less than t.

Answer 1

The correct answer is A. t, as lim sup tn = t indicates that the greatest limit point of the sequence tn is t, and therefore, there must be a subsequence of tn that converges to t.

Step-by-step explanation:

If a sequence tn has lim sup tn = t, there exists a subsequence converging to t. The correct answer to the question is A. t. The term lim sup or limit superior of a sequence refers to the greatest limit point of the sequence, which is the smallest value such that any number greater than it only occurs finitely many times in the sequence. Therefore, if the lim sup of a sequence is t, there must be a subsequence that converges to t, as this is the essence of what it means for t to be the lim sup. Subsequences that converge to any other value, like t+ϵ where ϵ > 0, or any value greater or lesser than t, are not guaranteed just by the definition of lim sup.