Theorem: n=n+1



Proof:

(n+1)^2 = n^2 + 2*n + 1



Bring 2n+1 to the left:

(n+1)^2 - (2n+1) = n^2



Substract n(2n+1) from both sides and factoring, we have:

(n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)



Adding 1/4(2n+1)^2 to both sides yields:

(n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2 = n^2 - n(2n+1) + 1/4(2n+1)^2



This may be written:

[ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2



Taking the square roots of both sides:

(n+1) - 1/2(2n+1) = n - 1/2(2n+1)



Add 1/2(2n+1) to both sides:

n+1 = n