- #1

Bashyboy

- 1,421

- 5

## Homework Statement

Let ##Q = \{(x_1,x_2,...,) \in \mathbb{R}^\omega ~|~ \lim x_n = 0 \}##. I would like to show this set is closed in the uniform topology, which is generated by the metric ##\rho(x,y) = \sup d(x_i,y_i)##, where ##d## is the standard bounded metric on ##\mathbb{R}##.

## Homework Equations

## The Attempt at a Solution

Let ##x = (x_n) \in \overline{Q} - Q##. Then ##\lim x_n = \ell \neq 0##, and WLOG take ##\ell > 0##. In my attempt, I mistakenly thought that ##\prod (x_n - \ell, x_n + \ell)## was a basis element for the product topology containing ##x##, concluding from this that it must be open in the uniform topology. From there I proved that if ##z = (z_n) \in \prod (x_n -\ell, x_n + \ell)##, then it cannot converge to ##0##, which gives the desired contradiction.

My question is, is ##\prod (x_n -\ell, x_n + \ell)## in fact open in the uniform topology?