Let X d be a metric space a Show that dP Q dP Q lessthanore

Let (X, d) be a metric space. a. Show that d(P, Q) - d(P, Q\') lessthanorequalto d(Q, Q\') for every P, Q, Q\' elementof X. b. Conclude that |d(P, Q) - d(P, Q\')| lessthanorequalto d(Q, Q\') for every P, Q, Q\' elementof X.

Solution

This is a simple proof.

Let P, Q & Q\' be three coplaner points. All three points are unique and they can either be collinear or non collinear.

Of they are non collinear then they will forum three vertices of a triangle P, Q&Q\'.

In any triangle we know that difference between two sides is always less than third side(by properties of sides of triangle).

When the points are collinear than

P, Q & Q\' will lie on same line and if Q is between P & Q\' then

d(PQ) +d(QQ\')=d(PQ\')

Thus,d(PQ) - d(PQ\')=d(QQ\')

Thus from both cases,

d(P, Q) - d(P, Q\') <=d(Q, Q\')

b). Clearly, measure of d(P, Q) - d(P, Q\') less than one equal to measures of (Q, Q\')

Thus,

|d(P, Q) - d(P, Q\') |<=|d(Q, Q\') |