To prove existence of a sequence $c_n$ such that $f(c_i)=f(c_j),\forall i,j \in \Bbb{N}$

Question

$f$ is continuous function and $I_n=[a_n,b_n]$ with infinite intersection which is non empty.Then there exist sequence $c_n$ such that $f(c_i)=f(c_j)$ for every $i,j \in N$

I am not able to start as I was thinking to use compactness property but not able to write. Any help will be appreciated