This is actually a generalized version I wrote of a homework question that intrigued me:
Let $f$ be continuous in $[0,1]$ and $\forall x\in [0,1], \ f(x)=f(1-x)$. If $r\in [0,1]$ and $1-r$ are the only sub limits of $a_{n}$ then $f(a_{n})$ converges.
(the original question states $r=\frac{1}{3}$)
I solved the original question using a method shown in class and tried to implement it to prove the generalized version (see my own answer). I was wondering if there's a better way.