# MATH10320 Mathematical Analysis

Mathematical Analysis is a large and important branch of modern mathematics with a long history. The core notion is that of a limit, which gives the "long-term" behaviour of some process, or computes what happens as one variable gets closer and closer to a given value. The limit notion is behind many other fundamental concepts, such as continuity, derivatives, integrals, and so on. Historically, such ideas were discussed and used since at least the 17th century (for example, in connection with mathematical physics or statistics). However, they were not precisely understood; the working definitions of the day typically involved somewhat vague ideas of "infinitely small" or "infinitely large" quantities, and these can lead to confusion.

We will follow in the footsteps of the pioneering mathematicians of the 19th century and formally define limits of sequences of real numbers. We will then use these definitions to rigorously deduce important properties of sequences, series and so on, which make up the bedrock of Real Analysis.

Topics investigated will include: The Completeness Axiom, Sequences, Series, Absolute and Conditional Convergence of Series,
Power series, Countability of sets, Continuity and properties of continuous functions, the Boundedness Theorem and the Intermediate Value Theorem.

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Curricular information is subject to change

Learning Outcomes:

On completion of this module the student should be able to
1. Compute the supremum and Infimum of sets of real numbers,
2. Prove or disprove elementary statements concerning the supremum and Infimum of sets of real numbers,
3. Show that certain sequences converge or diverge and determine their limits when they converge,
4. Use the Monotone Convergence Theorem to establish various properties of sequences and subsequences,
5. Test for convergence a wide range of series,
6. Be able to distinguish between the concepts of absolute and conditional convergence,
7. Determine the radius of convergence and interval of convergence of a power series
8. Be able to distinguish between countable and uncountable sets,
9. Prove or disprove elementary statements concerning continuous functions.

Student Effort Hours:
Student Effort Type Hours
Lectures

36

Tutorial

10

Autonomous Student Learning

65

Total

111

Approaches to Teaching and Learning:
Lectures, tutorials, problem-based learning
Requirements, Exclusions and Recommendations
Learning Recommendations:

At least H4 (or equivalent) in Leaving Certificate Mathematics

Module Requisites and Incompatibles
Incompatibles:
MST20040 - Analysis

Students must have completed MATH10350 or MATH10140. It is strongly recommended that students complete MATH10140 Numbers and Functions before taking this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade

Not yet recorded.

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour