Module: Introduction to Analysis

Latest Module Specifications

Current Academic Year 2025 - 2026

Module Title	Introduction to Analysis
Module Code	MTH1054 (ITS: MS323)
Faculty	Mathematical Sciences	School	Science & Health
NFQ level	8	Credit Rating	5

Description

This module introduces students to the formal and rigorous approach to mathematics which underpins mathematical analysis. The students will develop the skills necessary to make the transition from a formulaic understanding of mathematics to constructing their own formal mathematical arguments, and to promote advanced mathematical thinking through the use of guided inquiry and example generation.

Learning Outcomes

1. interpret the formal mathematical definitions and statements which arise in analysis.
2. classify and describe the main components of the definitions or statements, and the motivation behind them.
3. give examples or counterexamples of important phenomena which are studied in mathematical analysis.
4. critique and explain the logical steps which are required to apply definitions or theorems to the phenomena which occur in mathematical analysis.
5. critique and explain the main logical arguments which occur in the proofs of a selection of theorems.
6. calculate important quantities which arise in mathematical analysis e.g. bounds of sets or sequences, convergence of sequences or series.

*Type*	*Hours*	*Description*
Workload	Full time hours per semester
Tutorial	12	Discussion and feedback on weekly assignements
Lecture	24	Weekly lecture
Assignment Completion	22	Portfolio assignment
Independent Study	67	Recommended time for self-study
Total Workload: 125

Section Breakdown
CRN	20792	Part of Term	Semester 2
Coursework	0%	Examination Weight	0%
Grade Scale	40PASS	Pass Both Elements	Y
Resit Category	RC3	Best Mark	Y
Module Co-ordinator	Martin Friesen	Module Teacher

Type	Description	% of total	Assessment Date
Assessment Breakdown
Assignment	n/a	20%	As required
Formal Examination	End-of-Semester Final Examination	80%	End-of-Semester

Reassessment Requirement Type
Resit arrangements are explained by the following categories; RC1: A resit is available for both^* components of the module. RC2: No resit is available for a 100% coursework module. RC3: No resit is available for the coursework component where there is a coursework and summative examination element. ^* ‘Both’ is used in the context of the module having a coursework/summative examination split; where the module is 100% coursework, there will also be a resit of the assessment

Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Real numbers
Axiomatic definition of the reals, Inequalities, modulus function, triangle inequality, bounded sets, supremum and infimum

Sequences
bounded sequences, monotone sequences, convergent sequences, Cauchy sequences, Convergence theorems

Applications of sequences
Newton approximation for square roots, Exponential function and the logarithm, (Iterated fractions)

Application to Differential Calculus
Formulation of sequential continuity, Derivative, and Riemann integral in terms of sequences

Series
Geometric series, Telescoping series, Harmonic series, Leibnitz criterion, absolute convergence, Cauchy product, ratio test, root test

Power series
Convergence radius, Exponential function, Trigonometric functions, Differentiability, Integrability

Indicative Reading List

Books:

David Alexander Brannan: 0, A first course in mathematical analysis, Cambridge University Press, 978-0521684248
J. and P. Mikusinski: 0, An introduction to analysis, Wiley,
C. H. Edwards: 0, The historical development of the calculus, Springer-Verlag,

Articles:
None

Other Resources

None