$|a - c| \leq |a - b|+|b - c|$ holds for all real numbers $a$, $b$ and $c$.

This is in a way saying that the distance between $a$ and $c$ is less than or equal to the distance from $a$ to $b$ and then from $b$ to $c$.

Let $a$, $b$ and $c$ be real numbers. Let $x = a-b$ and $y = b-c$. Using the triangle inquality we know that

$$ \begin{align*} |x + y| &\leq |x| + |y| \\ |a - b + b - c| &\leq |a - b| + |b - c| \\ |a - c| &\leq |a - c| + |c - b|. \\ \end{align*} $$

$\blacksquare$