I am reading the wikipedia page on martingale difference sequences. $X_n$ is a martingale difference sequence with respect to a filtration $\mathcal{F}_n$ if it is adapted, integrable, and $E[X_n\mid \mathcal{F}_{n-1}]=0$.
In particular it says that if $Y_n$ is a martingale, then $X_n=Y_n-Y_{n-1}$ is a MDS. It does not make any mention of the converse. Is the converse true?
I think it is, because $Y_n=\sum\limits_{k=0}^{n} X_k$, so that $E[Y_n\mid \mathcal{F}_{n-1}]= Y_{n-1}+E[X_n\mid \mathcal{F}_{n-1}]=Y_{n-1}$.
PS: I know wikipedia isn't the best source. Could someone confirm whether or not I am correct?