MAT2101: Real Analysis I Pre-requisite

This is a pure mathematics course covering topics needed for analysis in physical, life and social science disciplines.  The topics covered include: The Real Number System R • Topology of Real Numbers; Sequences of Real Numbers; Function, Limits, Continuity and Uniform Continuity; Differentiability;  Series of Real Numbers;  Sequences and Series of Functions; and  Theory of Riemann Integrals.
 
Course Objectives   
On successful completion of this course unit, the learners should be able to: 

• Define Real Numbers R: And work with Real Numbers, Sate intervals of real numbers R, define Bounded intervals, Supremum and Infimum, state Completeness axiom, Archimedean principle, Bolzano Weierstrass Theorem and other important theorems. 
• Explain the concepts of Neighbourhood and prove theorems and apply them to interior point(s), exterior point(s), boundary point(s), open and closed sets, nested sets, accumulation point(s), closure, covering, connectedness and compactness. iii. Define a sequence, evaluate limit(s) of sequence and apply the ε-N definition of a limit of sequence to test convergence of sequence in regard to summation, subtraction, multiplication, division, and scalar multiplication of sequences, define a Cauchy sequence and sub-sequence, and state and prove the Cauchy general principle of convergence and Bolzano Weierstrass (version II) theorem. iv. Define series and their partial sums, state and prove Cauchy’s general principle of convergence of series, use a variety of test to test and/or prove for convergence of series among which include comparison tests, limit comparison test, absolute test, p-power test, root test, ratio test, and other general principles as applies to different types of series.  
• Apply the concepts of sequences and series of real numbers to define sequences and series of functions, state and prove the general theorems of point-wise convergence and uniform convergence of sequences of functions and apply these theorems to solve problems; state and prove Cauchy’s general principle of convergence of series of functions, state and prove the general theorems of tests of convergence of series of functions including the Weierstrass M-Test, and explore the consequences of general convergence. 
• Define limit of a function, give the ε-δ formal definition of a limit and continuity of function, and apply this definition to prove for convergence, continuity and uniform continuity, and apply general principle of convergence of sequences to function limits and continuity. 
• State and prove the general principle of continuity and differential, left and right derivatives, differentiation in an interval, stater and prove the theorems of the mean, the squeeze law and L’Hospital’s Rule, and applications to various techniques of differentiation.   
• Define and prove the general principles of Riemann integral, state the general notation of Riemann-Stieltjes Integral (RSI), state and prove the linear properties of RSI, and apply them to integration by parts. 

Expected Learning Outcome 
This course unit is meant: 

• To discuss the basic competence in the concepts, principles, and procedures of critical and logical mathematical analysis of real number system and functions with application to general theory of geometric differentiation and integration 
• To encourage orderliness, speed and accuracy in the presentation of mathematics. 
• To help learners acquire the skills of expression in proper mathematical language and using mathematical symbols correctly. 
• To provide instruction that contributes to the learners’ abilities to think critically and solve real life problems, to reason mathematically and apply computational skills. 
• To build a strong foundation in mathematical analysis as preparation for subsequent courses in pure mathematics and other sciences at advanced level.