HI, got an exam soon and need help understanding this question;

http://i97.photobucket.com/albums/l2...bbbbbbbb-1.jpg

So i know what a Cauchy sequence is i.e d(x_n,x_m)<=epsilon, but have no idea how to apply that to this problem.

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- May 22nd 2009, 01:02 PMpkrCauchy sequence
HI, got an exam soon and need help understanding this question;

http://i97.photobucket.com/albums/l2...bbbbbbbb-1.jpg

So i know what a Cauchy sequence is i.e d(x_n,x_m)<=epsilon, but have no idea how to apply that to this problem. - May 23rd 2009, 03:30 AMOpalg
This question is wrongly worded, because it claims that the functions are in the space , whereas in fact these functions are obviously discontinuous at x = 1/n.

Suppose we correct the question by redefining the functions to be continuous, say To show that they form a Cauchy sequence for the given metric, the easiest method is to show that they form a convergent sequence, with the limit function being the constant function g(x)=1.

The function is then zero except on the interval [0,1/n], where it is equal to 1– nx. Therefore as . Therefore the sequence is convergent and hence Cauchy.