### Understanding UCI SequencesLet us say that a sequence of real numbers \((a_n)\) is a UCI sequence if for every \( \epsilon > 0 \) there exists \( N \) such that \( n \geq N \) implies \( |a_{n+2} - a_n| \leq \epsilon \).#### Problem 1**Define the sequence**\[ a_n = \sum_{j=1}^{n} \frac{1}{\sqrt{j}} \]Show that \((a_n)\) is a UCI sequence.#### Problem 2**Prove the following statements:**1. Show that if \((a_n)\) is a Cauchy sequence then it is UCI.2. Is every UCI sequence Cauchy? Prove your answer.---### Explanation:A sequence \((a_n)\) is defined to be a UCI sequence if, for every \( \epsilon > 0 \), there exists an \( N \) such that for all \( n \geq N \), the difference between \( a_{n+2} \) and \( a_n \) is within \( \epsilon \).**1. Analysis of the given sequence:**The given sequence \((a_n)\) is defined as:\[ a_n = \sum_{j=1}^{n} \frac{1}{\sqrt{j}} \]To show that \((a_n)\) is a UCI sequence, we must demonstrate that for any \( \epsilon > 0 \), a point \( N \) can be found beyond which the difference \( |a_{n+2} - a_n| \leq \epsilon \) holds true for all \( n \geq N \).**2. Properties of UCI and Cauchy sequences:** - **Cauchy implies UCI:** Prove that if \((a_n)\) is a Cauchy sequence, then this sequence must be UCI. - **UCI implies Cauchy?:** Determine whether every UCI sequence is also a Cauchy sequence, and provide a proof or counterexample.By analyzing the definitions and properties, we delve into the behavior of sequences in terms of convergence and the relationship between Cauchy and UCI sequences. This exercise deepens our understanding of these mathematical concepts and their applications

Name: ### Understanding UCI SequencesLet us say that a sequence of real num
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: ### Understanding UCI SequencesLet us say that a sequence of real numbers \((a_n)\) is a UCI sequence if for every \( \epsilon > 0 \) there exists \( N \) such that \( n \geq N \) implies \( |a_{n+2} - a_n| \leq \epsilon \).#### Problem 1**Define th