If $d_1, d_2$ are metrics of $X$, is it true that $d_1 +d_2 $, $d_1 - d_2$, $d_1\cdot d_2$, $\sqrt d_1$ are metrics on $X$?
Here is my attempt:
If we take $d_1 = d_2 $ = standard metric on the real line, then $d_1\cdot d_2 = d_1^2$ is not a metric.
$d_1 - d_2$ may not be metric because it may not even be always non-negative.
But I am not sure about others. I need help.
Thanks for giving me time.