I am trying to show the existence of a unit vector fixed by a proper rotation in $\mathbb R^3$.
For a linear transformation $R$ in $\mathbb R^3$, to find a fixed vector, it is natural to consider eigenvector and therefore the characteristic polynomial $p(t)=det(R-tI)$.
But I do not how to proceed.
The key observations are $p(0)>0$ and $p(t)$ has negative leading coefficient(expand $p(t))$ as we are considering $\mathbb R^3$.
As $t\to\infty$, $p(t)\to -\infty$. So, we must have a $t_1>0$ such that $p(t)<0$. By intermediate value theorem, we have $p(t_0)=0$ for some $t_0>0$, which means $Ru=t_0u$ for some nonzero $u.$
Since this is a rotation $|Ru|=|u|=|t_0u|$, which implies $t_0=1$ . Done