Intermediate Value Theorem (IVT)

You've probably seen the problem where two people, $A$ and $B$, walk with different constant speeds $v_1$ and $v_2$, starting at different positions on the number line $p_1$ and $p_2$, and then meeting at an unknown time. We solved the problem by solving the linear equation $v_1 t + p_1 = v_2 t + p_2$, where $t$ is the unknown time.

Notice that this is equivalent to finding a zero for the function of their distance with each other $f(t) = (v_1 - v_2) t + (p_1 - p_2)$. Suppose that $v_1 < v_2$ and $p_1 > p_2$. Then $f(0) = p_1 - p_2 > 0$ and $$f(\frac{2(p_1 - p_2)}{v_2 - v_1}) = (v_1 - v_2) \frac{2 (p_1-p_2)}{v_2 - v_1} + (p_1 - p_2) = -(p_1 - p_2) < 0.$$ So based on IVT, there must be a time $t$ in $\left(0, \frac{2 (p_1-p_2)}{v_2-v_1}\right)$ such that they meet.