If each $\displaystyle X_{\alpha}$ is a Hausdorff space, then $\displaystyle \prod X_{\alpha}$ is a Hausdorff space in both the box and product top.

How do I go about proving this?

I know since $\displaystyle X_{\alpha}$ is Hausdorff, if for all $\displaystyle x_1,x_2\in X_{\alpha}$ where $\displaystyle x_1\neq x_2$ there exists neighborhoods $\displaystyle U_1 \ \text{and} \ U_2$ of $\displaystyle x_1 \ \text{and} \ x_2$ that are disjoint.