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Math Help - metrics - exam tomorrow!

  1. #1
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    metrics - exam tomorrow!

    Let d denote the usual metric on R and consider the metric e on R
    given by e(x, y) = |x − y|/(1 + |x − y|). (You may assume e is a
    metric.)

    In (R, e), show that B(0, 1) = R, and find B(0,1/3).

    Can someone please explain simply what I would have to do to get 4 marks?

    Thanks in advance!
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  2. #2
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    Have you drawn a graph of \frac{{\left| x \right|}}{{1 + \left| x \right|}}?
    It is essentially the same as that of the metric e, WHY?
    What are the upper and lower bounds?
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  3. #3
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by hunkydory19 View Post
    Let d denote the usual metric on R and consider the metric e on R
    given by e(x, y) = |x − y|/(1 + |x − y|). (You may assume e is a
    metric.)

    In (R, e), show that B(0, 1) = R, and find B(0,1/3).

    Can someone please explain simply what I would have to do to get 4 marks?

    Thanks in advance!
    Notice that \forall x, \forall y, e(x,y)<1. Thus, \forall~ x\in\mathbb{R}, e(0,x)<1 and it follows that B(0,1) will contain every point in \mathbb{R}.

    For the second part, y=0 and you need to find all the x such that \frac{|x|}{1+|x|}<\frac{1}{3}.

    3|x|<1+|x| \implies 2|x|<1 \implies |x|<\frac{1}{2}

    So B\left(0,\frac{1}{3}\right)=\left(-\frac{1}{2},\frac{1}{2}\right)
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