To determine the correct answer for this question, let's carefully analyze the definition and implications of having cluster points in a sequence of real numbers.
A cluster point (or limit point) of a sequence is a value that can be approached by the sequence as closely as desired, even though the sequence may not actually reach or converge to this value. This concept is rooted in the idea that within any neighborhood around a cluster point, there are infinitely many terms of the sequence.
Let's evaluate each option:
(a) It is convergent:
- Convergence of a sequence means that the sequence approaches a specific value as the index goes to infinity. Although the sequence having one or more cluster points could potentially mean that the sequence might converge to one of these points, it is not guaranteed. Having a cluster point does not necessarily imply the entire sequence converges.
(b) It is divergent:
- Divergence means that the sequence does not approach a finite limit as the index goes to infinity. Like with convergence, the presence of a cluster point does not necessarily mean the sequence is divergent either. The sequence could have multiple cluster points or cycle through values without settling, making this not a definitive conclusion.
(c) Limit exists:
- If the sequence has a cluster point, there could be a neighborhood around the cluster point containing infinite terms of the sequence, but it does not ensure that all terms of the sequence approach a single limit. Hence, the existence of a limit is not confirmed just with the presence of cluster points.
(d) Existence of limit not definite:
- This option indicates that the presence of cluster points does not provide a conclusive statement about the limit of the sequence. A sequence may have cluster points and still lack a definitive limit. Therefore, the presence of cluster points alone is insufficient to make a clear determination about the sequence's limit behavior.
Based on the analysis, the correct answer is:
(d) existence of limit not definite
