Final answer:
- If a sequence has a defined limit, it means that it approaches a certain value or extends indefinitely, and it is convergent if the limit is a real number. (the correct answer is C)
- If the limit inferior and limit superior of a sequence are equal, then the sequence is convergent, indicating it is approaching a specific value. (the correct answer is A)
Step-by-step explanation:
The question pertains to sequences and their limits, which is a concept in mathematical analysis. When we say that lim sn is defined for a sequence (sn), it means that as n goes to infinity, the terms of the sequence approach a certain value or extend towards infinity or negative infinity. If the limit exists as a real number or extends indefinitely to infinity or negative infinity, it is not sufficient to conclude that the sequence is bounded. However, it does indicate that the sequence is convergent if the limit is a real number or divergent if the limit is infinity or negative infinity.
Hence, the correct answer to the first question is C. The sequence is convergent if lim sn is a real number.
For the second question, if the lim inf sn (limit inferior) equals lim sup sn (limit superior), it signifies that the sequence has a specific limit from below and above. Therefore, the sequence converges to a particular number, and thus the sequence is bounded in this situation.
Hence, the correct answer is A. The sequence is convergent.