I encountered this in a book yesterday (de la Harpe, if you're interested) when I was a tad bored and I decided to "understand" the proof. It stated,

If we take

then

.

The first part of the proof is simple: take the unit square with

in its south-west corner. Then in even a worst case we see that the square is contained in the circle with center (0,0) and radius

(just applying Pythagoras to the two squares that we find ourselves with and summing). Applying the area of a circle formula, we get

for all

.

This is where I think we should stop. I mean, that is the statement in the Theorem, isn't it? However, the author then says "if the square corresponding to

touches the disc of radius

, then

; hence

for all

. Thus

for all

. Ends."

and I don't really see what we are doing continuing. I don't see what we are trying to prove, or how we get the absolute value sings.

Thanks in advance!