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.