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Math Help - complete space

  1. #1
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    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}

    Thanks in advance
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  2. #2
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    Re: complete space

    Okay, to start with, what is meant by "complete"? What is the precise definition?
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  3. #3
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    Re: complete space

    A metric space is complete if the Cauchy sequences converge in this space
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  4. #4
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    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 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?
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  5. #5
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    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.
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