Find the Error in the Following Proof of This "Theorem $P ( k ) \rightarrow P ( k + 1 )$

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 $P ( k ) \rightarrow P ( k + 1 )$ 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 ."