ELMO SL 2014 N2

Define the Fibanocci sequence recursively by $F_1=1$ , $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$ . Prove that for all integers $b,c>1$ , there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$ .

Proposed by Ryan Alweiss