I'm gonna give a slightly more mathematical answer here.
The Intermediate Value Theorom is defined as follows:
If $f(x)$ is a continuous function on $[a,b]$, for every $d$ between $f(a)$ and $f(b)$, there exists a $c$ between $a$ and $b$ so that $f(c) = d$.
Before we apply this theorom to the monk, let us define some values:
- Let $t$ be the time since the monk has begun an ascent or descent, in hours.
- Let $f(t)$ be the height of the monk going up the mountain.
- Let $g(t)$ be the height of the monk going down the mountain.
- Let the height the monk starts at be $0$, and the height of the mountain be $m$.
We are attempting to prove that there exists some $t$ in $[0,10]$, such that $f(t)=g(t)$.
We know that the height of the monk is a continuous function of time, both going up the mountain and back down, so we can add a couple more conditions:
- The domain of $f(t)$ and $g(t)$ are $t=[0,10]$.
- Both $f(t)$ and $g(t)$ are continous functions with the following properties:
Let us construct a function $d(t) = f(t) - g(t)$. This function is on the domain $t=[0,10]$ and the range $[-m,m]$, and remains continuous.
By applying the Intermediate Value Theorom, we know that for any $h$ in $[-m,m]$, there is a $t$ such that $d(t)=h$. If we take $h=0$, then we know that there is a $t$ such that $d(t)=0$.
Thus, we have proved that there is a $t$ such that $d(t)=f(t)-g(t)=0$, proving that there is a $t$ such that $f(t)=g(t)$. $\blacksquare$