Question

Prove shortly that |d(x,z)-d(y,z)|≤d(x,y) Give its Hint or prove it shortly.

Answer 1

Answer on Question #72875 – Math – Real Analysis

Question

Prove shortly that $|d(x,z)-d(y,z)| \leq d(x,y)$ Give its Hint or prove it shortly.

Solution

By the triangle inequality $d(x,z) \leq d(x,y) + d(y,z) \rightarrow$

$$\rightarrow d(x,z) - d(y,z) \leq d(x,y).$$

Also, by the triangle inequality $d(y,z) \leq d(y,x) + d(y,z) \rightarrow$

$$\rightarrow -d(y,x) \leq d(x,z) - d(y,z) \text{ or } d(x,z) - d(y,z) \geq -d(x,y)$$

because

$$d(x,y) = d(y,x)$$

It follows from (1) and (2) that

$$|d(x,z) - d(y,z)| \leq d(x,y).$$

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