Let $M_n$ be an $(\mathcal{F}_n)$ - martingale on $\left(\Omega,\mathcal{F},P\right)$, show that $$E[(M_n-M_m)M_k]=0,\qquad 0\leq k\leq m \leq n.$$
I want to know if this is correct:
By the multi-step property of a martingale we have that $$E[M_n|\mathcal{F}_m]=M_m\implies E[M_n-M_m|\mathcal{F}_m]=0,\qquad 0\leq m\leq n.$$
Multiplying my $M_k$ gives $$M_kE[M_n-M_m|\mathcal{F}_m]=0.$$ Since $\mathcal{F}_k\subseteq \mathcal{F}_m$, then $$E[(M_n-M_m)M_k|\mathcal{F}_m]=0$$ and taking expectations gives $$E[E[(M_n-M_m)M_k|\mathcal{F}_m]]=E[(M_n-M_m)M_k]=0.$$
Is this correct?
It looks fine to me. The only criticism I'd make is that your last line has the wrong order on the equalities, and $$E[(M_n-M_m)M_k]=E[E[(M_n-M_m)M_k|\mathcal{F}_m]]=0$$ reflects your logical argument better (to someone who knows the subject, your intent is clear, but to someone learning it, it might look like you're begging the question).
On a sidenote, I'm assuming your martingale is $L^2$. Otherwise, there's a problem when you first write $E[(M_n-M_m)M_k|\mathcal{F}_m]=0$