I have to prove that if $\displaystyle X_\alpha$ is Hausdorff for each $\displaystyle \alpha \in A$, then $\displaystyle X= \Pi_{\alpha \in A}X_\alpha$ is also Hausdorff.

I have proved this for the two dimensional case, and I know if A is finite then it would be true for all of them by induction, but it doesn't say anything about A being finite or infinite. Is this still true by induction if A is infinite?