A Really Weird Metric Space

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.