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Chapter 5. Limits and Continuity in Metric Spaces Limits of Sequences in Metric Spaces 5.1 Definition: Let (xn)n≥p be a sequence in a metric space X. We say that the sequence (xn)n≥p is bounded when the set {xn}n≥p is bounded, that is when there exists a ∈ X andr>0suchthatxn ∈B(a,r)forallindicesn≥p. For a ∈ X, we

Homework Assignment #4 Part 1: Timeshare classwork Follow the video “Timeshare solution 2” and submit your spreadsheet solution to the timeshare problem with color code consideration when: 1. Only same color allowed 2. Downward exchange is allowed Part 2: NLP classwork Follow the videos and submit your spreadsheet solution to 1. Snow removal probelm 2.

Chapter 1. Real Limits, Continuity and Differentiation Introduction Real analysis is similar to calculus with a strong emphasis placed on rigorous math- ematical proofs. In this first chapter, we shall prove some of the theorems, about limits, continuity and differentiation, which are normally omitted in calculus courses. We shall also introduce a few additional concepts.

Chapter 3. Sequences and Series of Functions Pointwise Convergence 3.1Definition: LetA⊆R,letf:A→R,andforeachintegern≥pletfn :A→R. We say that the sequence of functions (fn)n≥p converges pointwise to f on A, and we write fn → f pointwise on A, when lim fn(x) = f(x) for all x ∈ A, that is when for all n→∞ x ∈ A and

Chapter 5. Limits and Continuity in Metric Spaces Limits of Sequences in Metric Spaces 5.1 Definition: Let (xn)n≥p be a sequence in a metric space X. We say that the sequence (xn)n≥p is bounded when the set {xn}n≥p is bounded, that is when there exists a ∈ X andr>0suchthatxn ∈B(a,r)forallindicesn≥p. For a ∈ X, we

Chapter 1. Real Limits, Continuity and Differentiation Introduction Real analysis is similar to calculus with a strong emphasis placed on rigorous math- ematical proofs. In this first chapter, we shall prove some of the theorems, about limits, continuity and differentiation, which are normally omitted in calculus courses. We shall also introduce a few additional concepts.

Chapter 2. The Riemann Integral The Riemann Integral 2.1 Definition: A partition of the closed interval [a, b] is a set X = {x0, x1, · · · , xn} with a=x0 0sothatforeverypartitionXwith|X| 0 so that for every partition X of [a,b] with |X| < δ we 􏰈􏰈􏰓b􏰒n 􏰈􏰈 have 􏰈􏰈 f − f(ti)∆ix􏰈􏰈 <