Upper Bound for a Sum

For all $k$ between $1$ and $n$, we have that ${c_k}^2 \leq \sum \limits_{i=1}^n {c_i}^2$, therefore you get that $\sum \limits_{k=1}^n {b_k}^2 {c_k}^2 \leq \sum \limits_{k=1}^n {b_k}^2 \sum \limits_{k=1}^n {c_k}^2$, since all the ${b_k}^2$ are non negative. Now by Cauchy's inequality $(\sum \limits_{k=1}^n a_kb_kc_k)^2\leq \sum \limits_{k=1}^n {a_k}^2 \sum \limits_{k=1}^n (b_k c_k)^2 \leq \sum \limits_{k=1}^n {a_k}^2 \sum \limits_{k=1}^n b_k^2 \sum \limits_{k=1}^n c_k^2$.