Is the following true? If so, does anyone have a reference for a proof?
Claim: $\mathbb{E}[h(X_n, X_{n+1})(X_{n+1} - X_n)] = 0$ for any bounded, measurable $h$ if $X_n$ is a martingale.
I feel like it is probably true via monotone convergence arguments, however I have been struggling to find literature on the claim.
Suppose that $\left(X_n\right)_{n\geqslant 1}$ is a sequence of random variables such that $\mathbb{E}[h(X_n, X_{n+1})(X_{n+1} - X_n)] = 0$ for any bounded, measurable function $h$. Define the function $$ h=\colon \left(x,y\right)\mapsto \operatorname{sgn}\left(y-x\right), $$ where $\operatorname{sgn}\left(t\right)=1$ if $t\gt 0$, $-1$ if $t\lt 0$ and $\operatorname{sgn}\left(0\right)=0$. Since $t\cdot \operatorname{sgn}\left(t\right)=\left\lvert t\right\rvert$, we derive that $\mathbb E\left\lvert X_{n+1}-X_n\right\rvert=0$ hence $X_{n+1}=X_n$ almost surely.