Hello

I'm having trouble understanding something in Munkre's topology book. It's exercise 20.6.(a).

It says: Let $\displaystyle \rho$ be the uniform metric on $\displaystyle R^{\omega}$. Given $\displaystyle x=(x_1,x_2,...) \in R^{\omega}$ and given $\displaystyle 0< \epsilon <1$, let $\displaystyle U(x, \epsilon )=(x_1- \epsilon,x_1 + \epsilon) \times ... \times (x_n- \epsilon , x_n+ \epsilon) \times ...$ .

(a) Show that $\displaystyle U(x, \epsilon )$ is different from the $\displaystyle \epsilon$-ball $\displaystyle B_{\rho}(x, \epsilon )$

Now, I found a solution to this exercise that said that the point $\displaystyle (x_1+ (1/2) . \epsilon , x_2+ (2/3) . \epsilon , x_3 + (3/4) . \epsilon , ...)$ belonged to U, but not to the ball. How so?

Thanks