Let S be the subset of x-axis consisting of the union of the open interval of length 1/4 centered at 1/2, and the open intervals of length 1/16 centered at 1/4 and 3/4, the open intervals of length 1/64 centered at 1/8, 3/8, 5/8, and 7/8, and so forth. Let U = S x (0,1) be the union of the open rectangles of height 1, based on these intervals. Thus U is the union of one rectangle of area 1/4, two rectangles of area 1/16, four rectangles of area 1/64, ... some of which overlap.

show that the inner area of U is less than 1/2.

My first approach was to sum the areas of each rectange...

(ie) 1/4 + 2(1/8) + 4(1/6) ....

however this is incorrect because the rectangles intersect.

Any ideas/help?

thanks.

ahh.. stupid question, I see why none of you answered it.

$\displaystyle \sum_{k=0} 2^{k} (\tfrac{1}{4^{k+1}}) \leq \sum \tfrac{1}{2^{k+2}} $

Obviously this sum converges to 1/2.

Of course the rectangles intersect so we over estimate the real inner area, so clearly the inner area is less than 1/2.

HOWEVER the last part of this question is much more difficult.

Let V = R/U. Show that V is a closed set whose ourter area is bigger than 1/2.

help with the last part would be appreciated.