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- 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}. - A sequence (
*x*_{i}) in a metric space (*X*,*d*) is called a*Cauchy sequence*if:

Given > 0, there exists*N***N**such that*m*,*n*>*N**d*(*x*_{m},*x*_{n}) < .

Let (*f*_{n}) be the sequence of functions on [-1, 1] defined by

*f*_{n}(*x*) = 0 for*x*0,*f*_{n}(*x*) =*nx*for 0 <*x*<^{1}/_{n}and*f*_{n}(*x*) = 1 for*x*^{1}/_{n}.

Sketch the graphs of a typical pair of functions*f*_{m}and*f*_{n}and hence or otherwise prove that the sequence (*f*_{n}) is a Cauchy sequence in*C*[-1, 1] with the metric*d*_{1}.

Calculate the pointwise limit of the sequence (*f*_{n}) and hence prove that the sequence (*f*_{n}) is*not*a convergent sequence in*C*[-1, 1] under this metric. - Let
*f*be a map from a metric space*X*to a metric space*Y*. If the metric on*X*is the discrete metric, prove that*f*is continuous. If the metric on*Y*is discrete, is*f*necessarily continuous ? - This question tries to show you why the metric
*d*_{2}on a space of continuous functions is an easier one to work with than*d*_{1}or*d*_{}.

Find the best-fit straight-line through the origin to the function*x*^{2}in the metric*d*_{1}.

That is, 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

*x*-coordinate of the point*P*in terms of*a*.

Find the best-fit straight-line through the origin to the function*x*^{2}in the metric*d*_{}.

That is, 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.

Calculate*d*_{2}(*x*^{2},*ax*) and then use the usual calculus method for finding the maximal value of a function to find the best-fit straight-line through the origin to the function*x*^{2}in the metric*d*_{2}.

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