Let $\mathbb{R}^\omega$ denote the set of all the (infinite) sequences of real numbers.
Problem 2 (b), Sec. 25 in Munkres: Let $\mathbb{R}^\omega$ have the uniform topology. Then how to show that $x $ and $y$ are in the same component of $\mathbb{R}^\omega$ if and only if the sequence $$x-y = (x_1 - y_1, x_2 - y_2, x_3 - y_3, \ldots)$$ is bounded?
As I found the answer here..Probs. 2 (b) and 2 (c), Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Components in the uniform and box topologies. but I'm not getting this in my head.....
PLiz help me,,,