Let's assume all random variables here take positive values. All RVs will be represented in capital letters and constants in lower case. First, we know that $$\big| \mathbb{E}[A] - \mathbb{E}[B] \big| \le k$$
Now we also know that for RVs $M$ and $N$, $\mathbb{E}[M] = 1$ and $\mathbb{E}[N]=1$. Also $M$ and $N$ are bounded. I am trying to show (or find a counter-example) that $$\big| \mathbb{E}[AM] - \mathbb{E}[BN] \big| \le kc$$ In other words $\big| \mathbb{E}[AM] - \mathbb{E}[BN] \big|$ is on the order of $\big| \mathbb{E}[A] - \mathbb{E}[B] \big|$.
So far the most promising direction seems to be use Hölder's inequality where we have that $$\mathbb{E}[|AM|]\le\mathbb{E}[|A|]\sup|M|\le \mathbb{E}[|A|]\cdot h$$ and similarly $$\mathbb{E}[|BN|]\le\mathbb{E}[|B|]\sup|N|\le \mathbb{E}[|B|]\cdot l$$
However, the subtraction is throwing things off with my straight-forward calculation. I was hoping someone had a better idea for how to progress, or of course a counter-example?