Showing union of open sets is an open set?

cookiesyum

1. The problem statement, all variables and given/known data

Let U_n = {all p = (x, y) with |p - (0, n)| < n}. Show that the union of all the open sets U_n, for n = 1, 2, 3, ..., is the open upper half plane.

2. Relevant equations



3. The attempt at a solution

U_n describes points p whose distance from a set point on the vertical axis is smaller than the height of that point that is on the vertical axis. When you move the vertical axis point up and down, and combine all the sets of points created, you'll get the upper half plane not including the horizontal axis. I can picture it, but how do I begin to show it?

Dick

Pick a point in the upper half plane p=(x,y) with y>0. Can't you figure out a way to find an n large enough the p is in the circle centered at (0,n) with radius n? This really isn't conceptually hard. n can be as large as you like. Figure out the intersection of the circle with the vertical line through the point. Make it less than y.