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 v be a metric v : \mathbb{R}\times\mathbb{R} \to \mathbb{R} defined by

v(x,y) = Min(1, |x - y|)

We have shown in class that this generates the usual topology on \mathbb{R}.

Then, we will let p be a metric p : \mathbb{R}^{\infty}\times\mathbb{R}^{\infty}\to \mathbb{R} defined by

p(\bold{x},\bold{y}) = Sup\left\{\frac{v(x_j,y_j)}{j} : j \in \mathbb{N}\right\}

The problem is this:

Consider the set \mathbb{R}^{\infty} under the product topology. For each \bold{x}\in\mathbb{R}^{\infty}, let U = \prod\{X_j : j\in\mathbb{Z}^+\} be a basic open product containing \bold{x}. For each index j such that X_j \neq \mathbb{R}, let r_j be such that the segment (x - r_j,x + r_j)\subseteq X_j. Let r be the smallest of the quotients \frac{r_j}{j} and prove that the basic open B_r(\bold{x}) under the metric p is contained in U.

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.