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- Prove that the metrics
*d*_{1},*d*_{2}and*d*_{}on**R**^{2}do indeed satisfy the triangle inequality. What about the corresponding metrics on**R**^{n}? - Define a metric on the unit circle in
**R**^{2}by*d*(*X*,*Y*) = the length of the arc*XY*. Hence or otherwise prove that the formula*d*(*x*,*y*) =*min*{ |*x*-*y*|, 1 - |*x*-*y*| } defines a metric on the open unit interval (0, 1). Does the same formula define a metric on the closed interval [0, 1] ? - Let
*p*be a prime number. Define a function*f*on the positive integers**N**by the formula

*f*(*n*) =*p*^{k}where this is the largest power of*p*dividing*n*and a function*g*on the rationals**Q**by*g*(*a*/*b*) =*f*(|*a*|) /*f*(|*b*|) so that*g*(*r*) is the "largest power of*p*dividing*r*".

Prove that the function*d*given by*d*(*r*,*s*) = 1/*g*(|*r*-*s*|) defines a metric on**Q**.

[This is called the*p*-adic metric.]

Prove that in this metric the sequence (*p*,*p*^{2},*p*^{3}, ... ) converges to 0. - If
*P*and*Q*are any two points in**R**^{2}, prove that*d*_{1}(*P*,*Q*)*d*_{2}(*P*,*Q*)*d*_{}(*P*,*Q*).

Hence prove that if a sequence (**a**_{1},**a**_{2}, ...) in**R**^{2}is convergent in the metric*d*_{1}then it is also convergent in*d*_{2}and*d*_{}.

Prove also that*d*_{1}(*P*,*Q*) 2*d*_{2}(*P*,*Q*) and*d*_{2}(*P*,*Q*) 2*d*_{}(*P*,*Q*).

Deduce that if a sequence (**a**_{1},**a**_{2}, ...) in**R**^{2}is convergent in one of the metrics*d*_{1},*d*_{2},*d*_{}then it is convergent in all of them. - Which of the following real-valued functions on the open interval (0, 1) are continuous?

- Define a function
*f*on a real-number*x*by taking the decimal expansion of*x*(terminating in infinitely many 0's rather than infinitely many 9's if it is an exact decimal) and discarding the first, third, fifth and so on, decimal places.

So, for example,*f*( 0.1234) = 0.24,*f*(0.1415926536...) = 0.45256... . - Define a function
*g*on a real-number*x*by taking the decimal expansion of*x*and replacing 0's by 1's, replacing 1's by 2's and so on except that 9's are replaced by 0's.

So, for example,*g*(0.1298) = 0.23091111... (since the infinitely many 0's at the end all get replaced), g(0.1415926536...) = 0.2526037647... . - Define a function
*h*on a real-number*x*by taking the decimal expansion of*x*and replacing 0's by 9's, replacing 1's by 8's , 2's by 7's, 3's by 6's, 4's by 5's and vice-versa.

So, for example,*h*(0.1298) = 0.9701999999... = 0.9702 (since the infinitely many 0's at the end all get replaced),*h*(0.1415926536...) = 0.9594073463... .

[Hint: Observe that the sequence (0.49, 0.499, 0.4999, ...) converges to 0.5 and use the fact that continuous functions map convergent sequences to convergent sequences.] - Define a function

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