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- Define the metric
*d*_{2}on the space*C*[0,1] of continuous functions on [0,1] by*d*_{2}(*f*,*g*) =[(*f*(*x*) -*g*(*x*))^{2}*dx*]^{1/2}.

If*a*,*b*are fixed real numbers, calculate*d*_{2}(*x*^{2},*ax*+*b*). Use the usual method for finding the turning points of a function of two variables to find the values of*a*,*b*for which this distance is a minimum.

[This is the process of linear regression which is important in many areas of applied mathematics and statistics.] - Let (
*f*_{n}) be a sequence of continuous functions on a bounded interval, which satisfy |*f*_{n}(*x*)| ≤*M*_{n}for real numbers*M*_{n}and all*x*in the interval. If the*series**M*_{n}is convergent, prove that the partial sums of the series*f*_{n}(*x*) are uniformly convergent to a continuous function.

[Hint: Show that the sequence of partial sums of*f*_{n}(*x*) are a Cauchy sequence in*C*[0,1] and then use the result about completeness of this space.]

This result is known as the**Weierstrass**after the German mathematician Karl Weierstrass (1815 to 1897).*M*-test

In 1861 the German mathematician Bernhard Riemann (1826 to 1866) studied the function given by the series sin(*n*^{2}*x*)/*n*^{2}and tried to show that it was a continuous function whose derivative did not exist at any point. Use the last result to prove that this series does define a continuous function.

In fact this function*is*differentiable, though only at a countable subset of points. In 1872 Weierstrass defined the function*b*^{n}cos(*a*^{n}*x*) where 0 <*b*< 1 and*a*is an odd positive integer with*ab*> 1. He was able to prove that it is differentiable nowhere. Use the Weierstrass*M*-test to prove that it is continuous. This was the first such function to be discovered. - Define a function on the set of integers
**Z**by*d*(*m*,*n*) = 1/2^{r}if*m*≠*n*where 2^{r}is the largest power of 2 dividing*m*-*n*and*d*(*m*,*m*) = 0.

Prove that*d*is a metric on**Z**. This is called the 2*-adic metric*.

Prove that the sequence (1, 2, 4, 8, 16, ...) converges to 0 in this metric.

The mathematician Kurt Hensel (1861 - 1941, born in what was Königsberg in Germany and is now Kaliningrad in Russia) used ideas like this to prove results in Number Theory.

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