I am asked to prove or disprove the following:
Let $C\subseteq [0,1]$ be infinite. Then, there exists $a\in (0,1)$ s.t. $C\cap[a,1]$ is also infinite.
My attempt: Suppose that $C\cap[a,1]$ is finite for every $0<a<1$. Then, $C\cap[0,a)$ must be infinite. Consider an infinite sequence $c_1,c_2,c_3,...,c_n,...$ where $c_i\in C$ and $c_k<c_{k+1}$. By Archimedean property we can find $\frac{1}{n}=a<c_1$, such that $C\cap[a,1]$ is infinite, so we have a contradiction.
I'm not sure if I'm even allowed to do any of this, so I would appreciate if someone could guide me in the right direction.