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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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