Local martingale $\Leftrightarrow$ every component is a local martingale

Question

I want to prove the following corollary:

A $\mathbb K^d$-valued process $M$ is a local martingale if and only if every component process is a $\mathbb K$-valued local martingale.

($\mathbb K$ means $\mathbb R$ or $\mathbb C)$

I know that the statement holds true for martingales. I shall use the following exercise to prove the corollary:

Does anyone have an idea how to apply this exercise here?

Answer 1

For the "$\Rightarrow$"-direction you don't use this corollary. You just take the localizing sequence, then you know the stopped process is a martingale and use that you know that every component has to be also a martingale.

But for the "$\Leftarrow$"-direction you have to build a new localizing sequence of the localizing sequences of the components. This is where the corollary comes into play. You take for each $n$ the minimum of the localizing stoppingtimes of all components. Now the corollary tells you that this new stopping time resp. sequence is a localizing sequence for each component. Then you use again that the statement holds for martingales and you are done.