 If f, g are continuous functions on an interval [a, b], prove that d_{1}(f, g) ≤ (b  a)d_{∞}(f, g).
Hence prove that if a sequence (f_{n}) in C[a, b] converges to a function f in d_{∞} it also converges to f in d_{1}.
Solution to question 1
 Let f be a continuous function on the interval [0,1] for which f (0) = f (1). By considering the function h(x) = f (x)  f (x + ^{1}/_{2}) on a suitable interval, prove that there is some point x in [0, ^{1}/_{2}] for which f (x) = f (x + ^{1}/_{2}).
Deduce that there are two points opposite one another on the equator where the temperature is the same.
Solution to question 2
 Let f be a function from R to R for which there is a constant λ < 1 such that  f (x)  f (y)  ≤ λ x  y. Such a function is called a contraction mapping.
Choose a point x_{0} and define a sequence (a_{n}) by a_{1}= x_{0}, a_{n+1}= f (a_{n}) for n ≥ 1. Prove that this is a Cauchy sequence and that it has a limit a which is a fixed point of f. i.e. f (a) = a.
Prove that this fixed point is unique.
Solution to question 3
The next three questions try to show you why the metric d_{2} is an easier one to work with than d_{1} or d_{∞}.

Find the bestfit straightline through the origin to the function x^{2} in the metric d_{1}.
i.e find the value of a which minimises d_{1}(x^{2}, ax).
For this you have to minimise the shaded area.
Calculate this area by first finding the xcoordinate of the point P in terms of a.
Solution to question 4

Find the bestfit straightline through the origin to the function x^{2} in the metric d_{∞}.
i.e. find the value of a which minimises d_{∞}(x^{2}, ax).
For this you have to choose a so that the "vertical distance between the graphs" is a minimum.
From the picture, you can see that the best you can do is to adjust the value of a so that the two dark lines have equal length.
Solution to question 5
 Use the method of Exercises 9 Question 1 to find the bestfit straightline through the origin to the function x^{2} in the metric d_{2}.
Solution to question 6