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