During the 19th century, Mathematics underwent a number of ‘foundations crises' when it was realized that many of the key concepts upon which Calculus is based had no definition. Without proper definitions it was impossible to decide whether or not certain theorems were true, including

• there exists a function which is continuous everywhere and differentiable nowhere;
• Peano's theorem: given a rectangle, there exists a continuous curve which passes through every interior point of the rectangle.

This module will introduce you to logical quantifiers and to logical statements formed using these quantifiers. These will enable us to define certain key properties of sequences, series and functions such as boundedness, convergence and limit. You will see that certain ‘obvious' theorems are false, for example, a convergent series has the same sum if its terms are reordered.

You will learn a more axiomatic and rigorous approach to mathematics which is essential for studying the more advanced aspects of the subject with clarity and precision. You will learn to challenge yourself with the question, "If it's really so obvious, why can't I prove it?"

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

• make rigorous mathematical statements using logical quantifiers and form their negations;
• define upper and lower bounds, supremums and infimums, for sets, sequences and functions;
• explain the key differences between the reals and the rationals in terms of existence of supremums and infimums for bounded sets;
• define limits and convergence for sequences, prove convergence of sequences from first principles for simple examples; prove standard results on the `algebra of limits' and use them on examples;
• prove and apply the sandwich theorem;
• define monotonicity; prove and use the monotone convergence theorem;
• define convergence for series; prove and use some standard convergence tests;
• define injectivity, surjectivity and bijectivity for functions;
• define the limit of a function at a point and prove and use some standard related results;

27 - 50 minute lectures

5 - 50 minute tutorial classes

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills:

The ability to construct a rigorous mathematical argument using only the properties available from the hypotheses, without importing additional extraneous material from one's intuition.

Transferable Skills:

The ability to think and argue clearly and precisely.

The skill to present and explain mathematics to others and to work as a part of a team.

Formative assessment is by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.  

The in-course element of summative assessment is based on selected problems on the tutorial sheets.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• Real numbers
  • Natural, rational and real numbers. Examples of irrational numbers. Representation of real numbers as infinite decimals.

• Sequences
  • Convergence of sequences. Bounded and unbounded sequences. Formal definition of a limit and simple applications of this. Uniqueness of a limit. Boundedness of convergent sequences.
  • Simple uses of universal and existential quantifiers, including the negation of the definition of a limit.
  • Algebra of sequence limits. The sandwich theorem for convergent sequences.  Remarkable limits of sequences. Infinite limits.
  • Supremums and infinums. Existence of supremum for a set bounded above (without proof). Completeness and bounded monotonic sequences.

• Series
  • Convergence of series. Geometric series.  Tests for the Convergence of series with non-negative terms: comparison test and ratio test. Absolute and conditional convergence. Alternating series test.

• Limits of functions
  • Functional notation for real-valued functions, including composition and inverses. Surjective, injective and bijective maps on open and closed intervals.
  • Convergence of a function to a limit.  Definition of a limit and its illustration by examples.  Remarkable limits of functions. One sided and infinite limits. Theorems for sequences (Algebra of limits etc.) in the version for limits of a function at a point.

Please see Background Reading List for an indicative list.

Calculus, Spivak, M., Benjamin Cummings

Analysis 1, Tao, T., Hindustan Book Agency

Real Analysis, Howie, J.M., Springer

Mathematical Analysis, Binmore, K.G., Cambridge University Press

Numbers and functions: steps into analysis, Burn, R. P., Cambridge University Press