## Part 2

### New Skills

Formulating Maths
Introduction
Hypotheses &c
Implications
And & or
For all & there exists
Dependence

Proof
Introduction
Counter examples
Constructing proofs
Understanding
Experimentation
Precision
Examples
Assumptions
Induction
The End

Site map

Recall the three stages of problem solving:

(i)
understanding the problem (understanding the statement, and what is to be proved);
(ii)
experimentation (to find a method which looks likely to work);
(iii)
making the proof precise (making sure that the proof does work, and writing it out accurately and completely).
We discuss further examples highlighting these steps. These examples are arranged roughly in increasing order of sophistication. It is not expected that you will be able to solve all these problems, or even to understand all of them, on first reading of these notes. Do not despair: part of the purpose is to illustrate the need to find out what the statements mean before you start to prove them.

How to proceed:

First, read through the examples (below). It is likely that you will understand the statements of some of them (you may even be familiar with some of the results).

For each of those examples:

Try to solve the problem

• If you succeed
• write out a careful solution of your own
• and then compare it with the solution (click on the `solution' link corresponding to your problem)
• If you do not understand the statement of the problem
• then read the discussion of the example (click on the relevant `understanding' link)
• If you still can not solve the problem
• then read the treatment of the example (click on `solution').

The solutions have two parts: an experimentation stage and a precise write up of the proof.

Example 1. Let a, b, c and d be positive real numbers. Prove that

 a+b+c+d 4 ³ (abcd)1/4.

Understanding    Example 1.       Solution     of Example 1.

Example 2. Let n be a positive integer, and suppose that 2n-1 is prime. Prove that n is prime.

Understanding    Example 2.       Solution     of Example 2.

Example 3. Let n ³ 1. Prove that (1 - x/n )n £ e-x whenever 0 £ x £ n.

Understanding    Example 3.       Solution     of Example 3.

Example 4. Show that |sinx - siny| £ |x-y| whenever x,y Î R.

Understanding    Example 4.       Solution     of Example 4.

Example 5. Let f: Q ® R be a function such that f(x+y) = f(x)+f(y) for all x,y Î Q. Prove that f([m/n]) = [m/n]f(1) for all integers m and all strictly positive integers n.

Understanding    Example 5.       Solution     of Example 5.

Example 6. Let (an)n ³ 0 be a sequence of positive real numbers defined recursively by:

 a0 = 1,        an+1 = ____ Ö2+an . (*)
Prove that an converges to a limit as n®¥, and find the value of that limit.

Understanding    Example 6.       Solution     of Example 6.

Example 7. Let A and B be n ×n matrices. Prove that the traces of AB and BA are equal.

Understanding    Example 7.       Solution     of Example 7.

Design: Paul Gartside,
Content: Prof. C. Batty,
December 1999.
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