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

$\displaystyle v(x,y) = Min(1, |x - y|)$

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

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

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

The problem is this:

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

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.