Question

Assume that a sequence (sn) is not bounded above. Then there exists an increasing subsequence of (sn) ____.

A. Converging to a finite limit.
B. Diverging to infinity.
C. Oscillating.
D. With no limit.

Answer 1

If a sequence is not bounded above, there exists an increasing subsequence that diverges to infinity.

Step-by-step explanation:

Assuming that a sequence (sn) is not bounded above, it means that there is no limit to how large its terms can be. In this case, there exists an increasing subsequence of (sn) that diverges to infinity. This means that as we go further along the subsequence, the terms keep getting larger and larger without bound.

For example, consider the sequence (1, 2, 3, 4, 5, ...). This sequence is not bounded above because there is no largest term. If we take the subsequence of even numbers (2, 4, 6, 8, ...), we can see that it is an increasing subsequence that diverges to infinity.