1. ## complete space

Hi,
I need help to prove that the space of sequences of natural numbers X={xk} endowed with the following metric is complete:

d(X,Y)=1/min{k:xk is different from yk}

2. ## Re: complete space

Okay, to start with, what is meant by "complete"? What is the precise definition?

3. ## Re: complete space

A metric space is complete if the Cauchy sequences converge in this space

4. ## Re: complete space

Good. Now start by looking at simple cases. In the case that {x}= {1, 3, 2, 4 ....} and {y}= {1, 3, 4, 3, ...} d(x,y)= 1/3 because the third place is the first in which they are different.
What would it mean it $\displaystyle d(x, y)= 0$?

Notice that here the members of the metric space are sequences of real numbers so the "sequences" referred to in that definition are sequence of sequences!

If x and y are such sequences of different sequences, and d(x, y) goes to 0 what does that tell you about the sequences in x and y?

5. ## Re: complete space

The two sequences must be equal ,arn't they?So the convergent sequences are the eventually constant?That is {Xn}={{xn,k}:k=1,2,...} is eventually X={xk} for large enough n.