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
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.
Introduction
