Show that if d is a metric for X, thenis a bounded metric that gives the topology of X. [H...

Show that if d is a metric for X, then

is a bounded metric that gives the topology of X. [Hint: If f(x) = x/(1 +x) for x > 0, use the mean-value theorem to show that f(a + b) - f (b) ≤ f(a).]