union of two inequalities

Question

I would like to see if for two functions if it was possible to prove that for three functions $f, g, h$ that the union of

$| f -g| \geq t$ and $|g -h| \geq r$ where $ t,r \geq 0 $ would be equal to $|f -h| \geq t+r$

Answer 1

It is false. Consider the constant functions $f(x) \equiv 5, g(x) \equiv 3, h(x) \equiv 4$.

$|f-g| = |5-3| \ge 2$

$|g-h| = |3-4| \ge 1$

But $|f-h| = 1 < 2 + 1 = 3$