$X_1,X_2,\ldots$ are independent iid random variables, $Y_1,Y_2,...,Y_n$ (iid random variables) are independent from $X_{n+1},X_{n+2}...$ for every n.
$\mathcal{F}_{n}$ is a generated sigma algebra where $\mathcal{F}_{n}=\sigma(X_{1},...,X_{n},Y_{1},...,Y_{n})$ and $Z_{n}=\sum_{k=1}^{n}X_{k}Y_{k}$. We also know $\mathbb{E}(X_{n})=0$ and $\mathbb{E}(\left|X_{n}Y_{n}\right|)<\infty$ for every n.
How can I prove $(Z_{n},\mathcal{F}_{n})$ is a martingale?
Here is my results so far:
$$ \begin{split} E(Z_{n+1}\mid\mathcal{F}_{n}) &= E(\sum_{k=1}^{n+1}X_{k}Y_{k}\mid\mathcal{F}_{n})\\ &= \sum_{k=1}^{n+1} E(X_kY_k\mid\mathcal{F}_n) \\ &= \sum_{k=1}^{n}X_{k}Y_{k}+E(X_{n+1}Y_{n+1}\mid\mathcal{F}_{n})\\ &= Z_{n}+E(X_{n+1}Y_{n+1}\mid\mathcal{F}_{n}) \end{split} $$
I know I should prove somehow $E(X_{n+1}Y_{n+1}\mid\mathcal{F}_{n})=0$. I tried Law of total expectation but I didn't see how it would have helped. I know I should use the fact that $E(X_{n})=0$ and I should use the independence of the random variables but I've just stopped here. Can someone help me? Is it a good solution so far anyway?
Let $(X_i)_{i\geqslant 1}$ be an i.i.d. centered, square integrable sequence and let $Y_i=X_i$. Then all the conditions are fulfilled but $$ \mathbb E\left[X_{n+1}Y_{n+1}\mid\mathcal F_n\right]= \mathbb E\left[X_{n+1}^2\mid\sigma\left(X_1,\dots,X_n\right)\right]=\mathbb E\left[X_{n+1}^2\right]. $$