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Metric space and triangle inequality.

Prove that in a metric space, if C lies between A and B and O is any other point, then OC<=OA + OB. (Hint make 3 applications of the triangle inequality)

Triangle inequality: For triangle ABC AB+BC=>AC

Solution Preview

Cases when C lies on the line joining AB are trivial and can
be obtained just adding the 3 equations you wrote....
and using the result that AC+BC=AB here....

The general solution is to construct a rectangle around the line joining A and B. Draw lines parallel to x and y axis from A and B.

Since C is between A and B,C lies in this rectangle.....
If O is ...

Solution Summary

The triangle inequality is used to prove another inequality.