Chapter 2 introduces sequences, series, and several theorems for proving the convergence or divergence for each. However, the theorems cannot be used interchangeably for sequences and series. This Muddiest Point post will distinguish between some theorems for sequences and series. Convergence Theorems for Sequences The Monotone Convergence Theorem: If a sequence is monotone and bounded, […]
We started class on Thursday, as we will every Thursday from now on, with a situation for us to brainstorm what we actually know about the situation and what we don’t know, but wish we did. In this situation, we had a sequence whose even entries are bounded by 10. After sharing our ideas, we […]
By Zehao Dong and Zihan Hu 1 Introduction In Chapter 2, we studied the definition of sequence and the convergence of a sequence. Topological spaces provide a general framework for the study of convergence. However, instead of a distance function, we can think of the basic structure on a topological space as a collection of […]
We started class today by presenting challenge problems. We reviewed ways to solve problems 1a,b,c, 2 a,b, and 3 a,b. We also proved several theorems today. Theorem 2.3.2 says that every convergent sequence is bounded. Using the definition of convergence, we are able to prove this theorem in only a few lines. Theorem 2.3.3 is […]
In this post, I will discuss the following definitions we covered in Thursday’s (9/7) class: Chapter 1: Theorem 1.5.6 Chapter 2: Sequence, Convergence of a Sequence, Theorem 2.2.7 (Uniqueness of Limits) and Bounded Theorem 1.5.6 (i) Q ∈ N (Q is countable) (ii) R is uncountable (Proof Template) (i) Set A1 = {0} and for […]
Today we started class by discussing one final topic from Ch 1: Cardinality. Cardinality is a term used to refer to the size of a set. In discussing cardinality, we reviewed what it means for two sets to have a 1-1 correspondence, which can be defined as a function that is both 1-1 and onto. […]
By Grace Conway and Greg Bischoff Introduction The real numbers can be divided into many different categories to help facilitate different discussions of real numbers for different contexts. Usually in class we divide the real line into the rational and irrational numbers, but we don’t have to do it that way. We can discuss the […]
In Tuesday’s class we covered some challenge questions and some definitions and in this blog I will discuss about the following definitions that we addressed in class: Supremum and maximum, Theorem 1.4.2(Archimedean Property), Theorem 1.4.3(Density of Q in R) and Theorem 1.4.5 (There exists a real number α ∈ R satisfying α^2 = 2). 1.Supremum […]
In today’s class, we firstly presented some challenge problems. Among those interesting discussions which impressed me the most is the first challenge problem. This question needs us to apply knowledge of supremum and maximum. These 2 definitions are similar but absolutely different. Supremum is the least upper bound of a set but does not necessarily […]
You must be logged in to post a comment.