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|Theorem: n=n+1Proof:(n+1)^2 = n^2 + 2*n + 1Bring 2n+1 to the left:(n+1)^2 - (2n+1) = n^2Substract 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)^2This may be written:[ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2Taking 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