Final answer:
A bounded sequence may or may not converge, as boundedness indicates that the sequence is within a certain range but does not dictate that it must approach a specific value.
Step-by-step explanation:
The question concerns whether a bounded sequence must converge. The answer to this question is: b) The sequence may or may not converge. A sequence is bounded if there is a real number M such that every term of the sequence is within the interval −M and M. However, being bounded does not guarantee convergence. For a sequence to converge, it must approach a specific value as the number of terms goes to infinity.
A classic counterexample of a bounded but non-converging sequence is the sequence of ∑ and −1 alternating, which clearly does not approach any single value, demonstrating that boundedness alone is not sufficient for convergence. On the other hand, a sequence like 1/n, which is also bounded, does converge to zero. Thus, the boundedness of a sequence indicates that it’s contained within a specific range but does not necessitate convergence.