>>13547598We are, but you clearly don't have enough mathematical maturity to tackle this book. You keep posting questions from this book every day, filling up the catalog with your threads. And your questions are very simple, things that anyone with basic mathematical maturity could quickly answer. That's why I say you need to drop this book and learn how mathematical proofs work.
You said "x is any number between a and b, so it's continuous" demonstrating you don't understand what continuous means. Continuous only applies to functions not to numbers. You also demonstrate severe lack of reading comprehension. You ask where the sequence x_n comes from, but the author literally explains this in the previous sentence to the one you quoted.
>if F is not bounded, then for every positive integer n we can find a number x_n in the interval such that |f(x_n)|>n.Frankly I am amazed you got to chapter 4 of Rudin. I suspect you skipped a lot of material, and am almost certain you skipped most of the exercises. This is not how mathematics is learned. The chapter is clearly too complicated for you, so either go back to the first chapter and read it properly or drop Rudin and read something easier.
