MTH 207 — Real Analysis: Lecture Note

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LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Series 1: Bounds and Convergence of Sequences Series Outline Part 1.1 Bounds of Real Numbers 1.1.1 Definition of bounds 1.1.2 Upper and lower bounds 1.1.3 Supremum and infimum Part 1.2 Convergence of Sequences 1.2.1 Definition of sequence and limit 1.2.2 Convergent and divergent sequences 1.2.3 Examples of convergent sequences Part 1.3 Properties of Convergent Sequences 1.3.1 Uniqueness of limits 1.3.2 Algebra of limits 1.3.3 Order properties of limits Part 1.4 Applications of Convergence 1.4.1 Approximation of real numbers 1.4.2 Use in analysis and calculus lntroduction Real numbers form the foundation of analysis, and understanding how sequences behave is one of the first steps in building this knowledge. Sequences appear naturally in mathematics, for example when approximating irrational numbers or studying infinite processes. The ideas of bounds and convergence help us to determine whether a sequence stabilises towards a particular value or continues without limit. This Series will therefore provide you with the essential tools to study sequences rigorously, which is crucial for later topics in Real Analysis. The aim of this Series is to equip you with the ability to define bounds of real numbers, explain the concept of convergence, and apply the properties of convergent sequences to practical problems in analysis.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS We shall commence this Series by examining the bounds of real numbers, including upper and lower bounds, as well as the important notions of supremum and infimum. Next, we will move on to the convergence of sequences, where we will define limits and distinguish between convergent and divergent sequences with examples. Subsequently, we will consider the properties of convergent sequences, such as uniqueness of limits, algebra of limits, and order properties. Finally, we will conclude by exploring applications of convergence, particularly in approximating real numbers and in laying the groundwork for calculus. Series Learning Outcomes By the end of Series 1: Bounds and Convergence of Sequences, you should be able to: SLO 1.1 define bounds of real numbers and distinguish between upper bounds, lower bounds, supremum, and infimum, SLO 1.2 explain the concept of a sequence and its limit, and differentiate between convergent and divergent sequences, SLO 1.3 analyse the properties of convergent sequences, including uniqueness of limits, algebra of limits, and order properties, SLO 1.4 apply the concept of convergence to approximate real numbers and to support further study in analysis and calculus. 1.1 Bounds Of Real Numbers 1.1.1 Definition of bounds A bound is a value that restricts how large or small the elements of a set or sequence can be. If a set of numbers does not exceed a certain value, we say that value is a bound. Bounds are important because they help us describe the limits within which numbers lie, and they provide a foundation for understanding convergence. Fill in the blank: A value that restricts how large or small the elements of a set can be is called a __________________.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Illustration of a bounded set on the real number line 1.1.2 Upper and lower bounds An upper bound of a set is a number that is greater than or equal to every element in the set. A lower bound is a number that is less than or equal to every element in the set. For example, in the set 3, the number 3 is an upper bound, while 1 is a lower bound. Fill in the blank: A number that is greater than or equal to every element in a set is called an __________________.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.1.3 Supremum and infimum The supremum of a set is the least upper bound, meaning it is the smallest number that is greater than or equal to all elements of the set. The infimum is the greatest lower bound, meaning it is the largest number that is less than or equal to all elements of the set. These concepts are crucial because not all sets have maximum or minimum values, but they always have a supremum and infimum if they are bounded. Fill in the blank: The least upper bound of a set is called the __________________. Key differences between maximum/minimum and supremum/infimum Feature Maximum / Minimum Supremum / Infimum Definition The largest (max) or smallest (min) value within the set. The least upper bound (sup) or greatest lower bound (inf). Membership Must be an element of the set. May or may not be an element of the set. Existence Does not always exist, even for bounded sets (e.g., open intervals). Always exists for any non-empty set bounded above/below (Completeness Axiom). Relationship If a maximum exists, it is also the supremum. If the supremum is in the set, it is also the maximum. Pilot Questions 1.1 A value that restricts how large or small the elements of a set can be is called a: A. Limit B. Bound C. Sequence D. Supremum (Linked to SLO 1.1) A number that is greater than or equal to every element in a set is called a: A. Lower bound B. Supremum C. Upper bound D. Infimum (Linked to SLO 1.1) The least upper bound of a set is known as the: A. Maximum B. Supremum C. Limit D. Bound (Linked to SLO 1.1)

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS The greatest lower bound of a set is called the: A. Infimum B. Minimum C. Bound D. Upper bound (Linked to SLO 1.1) Which of the following statements is true? A. Every set has a maximum and minimum. B. Every bounded set has a supremum and infimum. C. Supremum and maximum are always the same. D. Infimum and minimum are always the same. (Linked to SLO 1.1) 1.2 Convergence Of Sequences 1.2.1 Definition of sequence and limit A sequence is an ordered list of numbers, often defined by a rule or formula. For example, the sequence an=1nproduces the terms 11,12,13,…. A limit of a sequence is the value that the terms of the sequence approach as nbecomes very large. If the terms get closer and closer to a single number, we say the sequence converges to that limit. Fill in the blank: A sequence that approaches a single value as nbecomes very large is said to __________________.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Graph of the sequence an=1napproaching the limit 0 AI prompt: 1.2.2 Convergent and divergent sequences A convergent sequence is one whose terms approach a finite limit. For example the sequence an=1nconverges to 0. A divergent sequence is one that does not approach any finite value. For instance, the sequence an=ndiverges because its terms grow without bound. Fill in the blank: A sequence whose terms grow without bound is called a __________________ sequence.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS comparison of a convergent sequence and a divergent sequence 1.2.3 Examples of convergent sequences Examples help to clarify the concept of convergence. Consider the sequence an=12n. As nincreases, the terms become smaller and approach 0. Another example is an=nn+1, which approaches 1 as ngrows. These examples show that convergence can occur towards different values, not just 0. Fill in the blank: The sequence an=nn+1converges to __________________. Pilot Questions 1.2

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