There is a nice elementary topology problem (proposition) that is often missing from the introductory books on the topic.
PROBLEM. Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a continuous self-map of the circle of degree $\deg(\varphi)=d$. Then $\varphi$ has at least $|d-1|$ fixed points. (For example, if $\varphi$ is an orientation reversing homeomorphism, then it has at least 2 fixed points - the 'two monks walking in opposite directions' problem.)
Its place should be just after the notion of degree, fundamental group etc. In my opinion, it is a very good exercise, as it combines basic notions, such as degree, fixed point, $\pi_{1}(\mathbb{S}^{1})$ and has useful applications. Paradoxically, I don't see it in the appropriate place ("degree", "fundamental group"), but in the more heavy advanced context of Nielsen theory. Nielsen theory, in its turn, is often missing from elementary topology books. The available to me ones are dealing with almost one and the same list of problems (nice, indeed), but this one seems not to be present there.
So my question is: Does anyone know a good elementary proof of this problem/proposition (without referring to advanced things such as Nielsen index theory or so). Any references are welcome as well. Thanks in advance.