Let $(a_0,a_1,a_2,...)$ and $(b_0,b_1,b_2,...)$ be such sequences of non-negative real numbers, that for every integer $i\geqslant 1$ holds $a_i^2\leqslant a_{i-1}a_{i+1}$ and $b_i^2\leqslant b_{i-1}b_{i+1}$. Define sequence $c_0,c_1,c_2,...$ as $$c_0=a_0b_0, \; c_n=\sum_{i=0}^{n} {{n}\choose{i}} a_ib_{n-i}.$$Prove that for every integer $k\geqslant 1$ holds $c_{k}^2\leqslant c_{k-1}c_{k+1}$.