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    Just starting metrics

    Consider the set S of all real numbers, but with a new distance function defined:

    d(x,y) = |\frac{x}{1 + x} - \frac{y}{1 + y}|

    Add two new points, +\infty and -\infty to the set S. call the resulting set \bar{S} = S \cup \{+\infty,-\infty\}.

    Now extend d to \bar{S} by setting d(x,+\infty) = |\frac{x}{1 + x}-1|, d(x,-\infty) = |\frac{x}{1 + x}-1| and d(+\infty,-\infty) = 2.

    (a) Prove: a sequence of reals x_n \in \bar{S} converges to +\infty \iff x_n \rightarrow \infty.

    (b) Prove: The sequence s_n \in \bar{S} converges \iff it is d-Cauchy.

    Any guidance would be greatly appreciated. Thanks!
    Last edited by Anonymous1; November 16th 2009 at 07:08 PM.
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