Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.

b.If a sequence of positive numbers converges, then the sequence is decreasing.

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Recall the definition of a sequence converging: A sequence \( \{a_n\} \) converges to a limit \( L \) if for every \( \epsilon > 0 \), there exists an \( N \) such that for all \( n > N \), \( |a_n - L| < \epsilon \).

Understand what it means for a sequence to be decreasing: A sequence \( \{a_n\} \) is decreasing if \( a_{n+1} \leq a_n \) for all \( n \).

Analyze the statement: "If a sequence of positive numbers converges, then the sequence is decreasing." This implies that every convergent sequence of positive terms must be decreasing.

Consider a counterexample to test the statement: For instance, the sequence \( a_n = \frac{1}{n} \) is positive and converges to 0, and it is decreasing. However, the sequence \( a_n = \frac{1}{n} \) is decreasing, but what about a sequence like \( a_n = \frac{1}{n} \) for odd \( n \) and \( a_n = \frac{1}{n} + \frac{1}{n^2} \) for even \( n \)? This sequence is positive and converges to 0 but is not strictly decreasing.

Conclude that convergence of a positive sequence does not guarantee it is decreasing, so the statement is false. A convergent sequence can oscillate or increase at some points as long as it approaches the limit.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Convergence of a Sequence

A sequence converges if its terms approach a specific finite limit as the index goes to infinity. Convergence does not impose restrictions on the sequence's monotonicity; the terms can oscillate or increase before settling near the limit.

Monotonic Sequences

A sequence is decreasing if each term is less than or equal to the previous term. Monotonicity is a property describing the sequence's order, independent of whether it converges or not.

Counterexamples in Sequence Analysis

To disprove a statement about sequences, providing a counterexample—a sequence that meets the conditions but violates the conclusion—is effective. For instance, a convergent sequence of positive numbers that is not decreasing shows the statement is false.

Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.b.If a sequence of positive numbers converges, then the sequence is decreasing.