Find the Error in the Following Proof of This "Theorem $' n = n + 1 '$

Short Answer

Find the error in the following proof of this "theorem":
"Theorem: Every positive integer equals the next largest positive integer."
"Proof: Let P(n) be the proposition $' n = n + 1 '$
To show that $P ( k ) \rightarrow P ( k + 1 )$ assume that P(k) is true for some k , so that k=k+1 . Add 1 to both sides of this equation to obtain k+1=k+2 , which is P(k+1) . Therefore $P ( k ) \rightarrow P ( k + 1 )$ is true. Hence P(n) is true for all positive integers n . "