**Proposition 1:** For any space $X$ and an infinite cardinal $\kappa$, the product $X^{\kappa}$ is either meager or a Baire space.

Does anyone have any suggestions to demonstrate Proposition 1?

I was trying using topological games in the following way, suppose that $X^{\kappa}$ is not a Baire space, then $X^{\omega}$ is not a Baire space (*Theorem 2, Barely Baire Spaces - W. Fleissner and K. Kunen*), then, using the Banach-Mazur game, Player I has a winning strategy in $\textsf{BM}(X^{\omega})$. I was trying to show that Player I has a winning strategy in the game $\textsf{MB}(X^{\kappa})$, but unfortunately I still haven't got it.

Could someone tell me if I am on the right path for the proof of the proposition? Otherwise, someone has another idea for the proof.

Thanks