(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am stuck on a step from a simple proof in Gelfand's method of coordinates.

Here is a link to the part I am confused on. Pg. 44-45...

http://books.google.com/books?id=In...ether with an estimate of its error:"&f=false

It starts in the middle of page 44 and ends right below the part I have highlighted on page 45.

2. Relevant equations

pi*r^2 is area of a circle.

2*pi*r is the circumference of a circle.

diagonal of a 1 unit^2 square is √(2) units

distance formula is (x1-x2)^2 + (y1-y2)2 = d^2

3. The attempt at a solution

when it asked to prove that "the figure A_{n}lies entirely within the circle K_{n}'' and contains the circle k_{n}' entirely within itself," I came up with an attempt by making a triangle ABC with A at (0,0), B at (B_{x},0) and C at (0,C_{y}) and saying AB + AC < BC

next, using the distance formula, I changed AB to B_{x}^{2}, AC to C_{y}^{2}, and BC to B_{x}^{2}+ C_{y}^{2}... That would mean that B_{x}^{2}+ C_{y}^{2}< B_{x}^{2}+ C_{y}^{2}which is clearly impossible! Does that qualify as a proof?

So since K_{n}'' has a radius that is √(2) bigger than K_{n}, then if the upper right vertex of a unit square is within the circumference of K_{n}, then it must also be within the circumference of K_{n}''. Because in order for it to extend beyond the circumference of K_{n}'' while still remaining in the circumference of K_{n}, then it would have to take up a distance larger than √(2) which is not possible for a 1 unit square. Same goes for K_{n}' not being completely within A_{n}because that would mean that one part of a unit square was inside the circumference of K_{n}' while the upper right hand corner of the same square was out of K_{n}... Not possible unless it had a straight line distance within the unit square that was longer than √(2).

OOOkay. Now that we have that part out of the way (hopefully it's right), the part I'm confused on is how they got from all the lead up steps to their final conclusion of:

|N-pi*n| < 2*pi(√(2n) + 1)

... I think they are trying to quantify the difference between the unit squares in A_{n}and the actual area of circle K_{n}. So they set minimum and maximum boundaries (K_{n}' and K_{n}'' respectively) that can help out. K_{n}' tells us the smallest circular boundary where A_{n}will stay within and K_{n}'' gives us the largest circle that will still be completely filled by A_{n}.

But i'm stuck on how they got the right side of the above inequality... All help will be appreciated; sorry for the long post!

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# Number of integer solutions to x^2 + y^2 <= n? [simple proof]

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