Exercise 12

What is wrong with the following “proof” of the “fact” that $n+3 = n+7$ for all values of $n$ (besides of course that the thing it is claiming to prove is false)?

Let $P(n)$ be the statement that $n + 3 = n + 7$. We will prove that $P(n)$ is true for all $n \in \N$. Assume, for induction that $P(k)$ is true. That is, $k+3 = k+7$. We must show that $P(k+1)$ is true. Now since $k + 3 = k + 7$, add 1 to both sides. This gives $k + 3 + 1 = k + 7 + 1$. Regrouping $(k+1) + 3 = (k+1) + 7$. But this is simply $P(k+1)$. Thus by the principle of mathematical induction $P(n)$ is true for all $n \in \N$.