Prove $ \sum a_i b_i \sum c_i\leq \sum a_i \sum b_i c_i $ [closed]

Question

Suppose $a_i, b_i, c_i >0$, $a_i\leq c_i$ for all $i=1,2,\ldots,n$. Prove $$ \sum a_i b_i \sum c_i\leq \sum a_i \sum b_i c_i.$$

This question is a simpler version of my last question. If this question can be proved, my last question is also done.

Answer 1

This is not true in general, $b_1 = 1$, $b_2 = 2$, $a_1 = 1$, $a_2 = 1$, $c_1 = 3$ and $c_2 = 2$:

$$\sum a_i b_i \sum c_i = (1\times 1 + 1\times 2) \times (3 + 2) = 15$$

$$\sum a_i \sum b_i c_i = (1 + 1) \times (1 \times 3 + 2 \times 2) = 14$$