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.