What Is Wrong with the Following Proof That Every Positive $P ( n )$

Short Answer

What is wrong with the following proof that every positive integer equals the next larger positive integer? "Proof." Let $P ( n )$ be the proposition that $n = n + 1$ . Assume that $P ( k )$ is true, so that $k = k + 1$ . Add 1 to both sides of this equation to obtain $k + 1 = k + 2$ . Since this is the statement $P ( k + 1 )$ , it follows that $P ( n )$ is true for all positive integers $n$ .