Multiplying non-decreasing sequences

Question

Let $(a_n)$ and $(b_n)$ be non-decreasing sequences of positive terms (i.e. $a_n\gt0$ and $b_n\gt0$ for all $n\ge1$).

Prove that the sequence $(c_n)$ is non-decreasing, where $c_n=a_nb_n$ for all $n\ge1$.

I know that multiplying two null sequences gives a null sequence, but how can I show that doing the same for non-decreasing sequences, yields a non-decreasing sequence?

Answer 1

Since $a_{n+1} \ge a_n$, we have:

$$a_{n+1}b_{n+1} \ge a_nb_{n+1}$$

Since $b_{n+1} \ge b_n$, we have:

$$a_nb_{n+1} \ge a_n b_n$$

Therefore,

$$a_{n+1}b_{n+1} \ge a_nb_n$$ $$c_{n+1} \ge c_n$$