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Suppose the sequence is bounded. Determine if the sequence must converge. Provide a proof or a counterexample justifying your solution.

a) The sequence must converge.

b) The sequence may or may not converge.

c) The sequence must diverge.

d) Insufficient information to determine.

1 Answer

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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.

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