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    Real Anaylsis : Metrics Proof

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    Let d be a metric in X. Prove that p(x,y)=(d(x,y))/(1+d(x,y)) is also a metric in X.

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    Solution Preview

    From the condition, d is a metric in X. To show that p is a metric in X, we have to verify the followings.
    (1) p(x,y)>=0.
    Since d(x,y)>=0, then p(x,y)=d(x,y)/(1+d(x,y))>=0.
    (2) p(x,y)=0 if and only if x=y.
    If p(x,y)=d(x,y)/(1+d(x,y))=0, then d(x,y)=0, then x=y because d is a metric.

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