Question

Let {an}_n≥1 be a sequence of non-zero real numbers. Then which one of the following statements is true ?

• A. If \(\left\{\frac{a_{n+1}}{a_n} \right\}_{𝑛≥1}\) is a convergent sequence, then {an}_n≥1 is also a convergent sequence
• B. If {an}_n≥1 is a bounded sequence, then {an}_n≥1 is a convergent sequence
• C. If |a_n+2 - a_n+1| ≤ \(\frac{3}{4}\) |a_n+1 - a_n| for all n ≥ 1, then {a_n}_n≥1 is a Cauchy sequence
• D. If {an}_n≥1 is a Cauchy sequence, then {an}_n≥1 is also a Cauchy sequence

Answer 1

The Correct Option is C

The problem involves determining the true statement about a sequence of non-zero real numbers and involves concepts related to convergence and Cauchy sequences. Let us evaluate each given statement one by one:

1. Statement 1: If \(\left\{\frac{a_{n+1}}{a_n} \right\}_{𝑛≥1}\) is a convergent sequence, then \({a_n}_{n \geq 1}\) is also a convergent sequence.

This statement is not necessarily true. Convergence of the ratios does not imply convergence of the original sequence. The original sequence could still diverge even if the ratio sequence converges.

1. Statement 2: If \({a_n}_{n \geq 1}\) is a bounded sequence, then \({a_n}_{n \geq 1}\) is a convergent sequence.

A bounded sequence is not necessarily convergent. For example, the sequence \( (-1)^n \) is bounded but not convergent.

1. Statement 3: If \(|a_{n+2} - a_{n+1}| \leq \frac{3}{4} |a_{n+1} - a_{n}|\) for all \( n \geq 1 \), then \({a_n}_{n \geq 1}\) is a Cauchy sequence.

This statement is true. If the differences between successive terms are getting progressively smaller, eventually approaching zero, the sequence will demonstrate the Cauchy property. According to the given condition, each successive difference is reduced by a factor of 3/4, leading to conclusion that the sequence is Cauchy.

1. Statement 4: If \({a_n}_{n \geq 1}\) is a Cauchy sequence, then \({a_n}_{n \geq 1}\) is also a Cauchy sequence.

This statement is trivially true because it is stating that a Cauchy sequence remains a Cauchy sequence which is a tautology without providing any new information.

The correct answer is that if the condition \(|a_{n+2} - a_{n+1}| \leq \frac{3}{4} |a_{n+1} - a_{n}|\) holds for all \( n \geq 1 \), the sequence is a Cauchy sequence.