Let d1 and d2 be two metrices on X . Then $d(x,y)= d1(x,y)*d2(x,y)$ $x, y\in X$ is also metric on X?

Question

I'm able to solve three properties of metric such as

1. $d(x,y)\geq 0$ for all $x, y \in X $
2. $d(x,y)=0$ iff $x=y$
3. $d(x,y)= d(y,x)$ for all $x, y \in X $

But facing problem to solve triangle inequality. Please help me. Thanks in advance.

Answer 1

Certainly false. Try to show that the square of the usual metric on the real line is not a metric.

Answer 2

$|0-\frac12|^2 + |\frac12 - 1|^2 = \frac12 < 1 = |0-1|^2$ so $d_1 = d_2$ equal to the standard distance on $\Bbb R$ already gives a counterexample to the triangle inequality.