**Sequence and Limit Point Analysis**Consider the sequence defined by \( x_n = 1 - \frac{1}{n} \).- **Graphical Representation:** This sequence can be visualized on the real number line. As \( n \) increases, the value of \( x_n \) approaches 1. You would observe that for each successive term, the value gets closer to 1 but never actually equals 1.- **Open Set around the Limit Point:** The limit point for this sequence is 1. To define an open set \( U \) around this limit point, we can choose an open interval (for example, \( U = (0.9, 1.1) \)). For \( N = 3 \), all \( x_n \) for \( n \geq N \) are within this open interval \( U \). This implies that starting from \( n = 3 \) onwards, the terms are inside the interval \( (0.9, 1.1) \), indicating that the sequence eventually stays within any neighborhood around the limit point as \( n \) progresses.This task involves understanding how sequences behave as they progress and how to describe their convergence using limit points and neighborhoods in the context of real analysis.

Name: **Sequence and Limit Point Analysis**Consider the sequence defined by
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: **Sequence and Limit Point Analysis**Consider the sequence defined by \( x_n = 1 - \frac{1}{n} \).- **Graphical Representation:** This sequence can be visualized on the real number line. As \( n \) increases, the value of \( x_n \) approaches 1. Y