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.

... is part 3 of the definition of metric space (page 37) we have. ,. therefore. . (b.1). The triangle inequality also says that. ,. therefore. . (b.2). Inequalities...

... Problem 11 Metric space C_[a,b] is defined on ... With metric defined by equation (0.1) it means simultaneous ... C_[a,b] we use triangle inequalities to construct ...

... A real analysis for textbook metric spaces is examined. ... (b) If the distance between x and a is defined, then the space is metric, reverting to case (a). ...

... As an aside, the triangle inequality is an axiom of all metric spaces and all normed vector spaces, so I do not explect you have to prove it separately. ...

... week, months of the year, seasons, triangle, rectangle, square ... will arrange and describe objects in space by proximity ... of inch, foot, yard, and metric units of ...

... fact, from the triangle inequality for a vector space. one finds that as n and m go to infinity. or more generally. as n and m go to infinity in a metric space. ...