So, I've been asked to prove this. I've no idea where to start, would like to see what this problem is actually talking about.

First, let be a metric defined by

We have shown in class that this generates the usual topology on .

Then, we will let be a metric defined by

The problem is this:

Consider the set under the product topology. For each , let be a basic open product containing . For each index such that , let be such that the segment . Let be the smallest of the quotients and prove that the basic open under the metric is contained in .

There's a lot of material in here. I just need some help getting started, or at least seeing what would be sufficient to show.