This is quite an advantage: we now have a stronger inductive hypothesis. We can assume that \(P(1)\text{,}\) \(P(2)\text{,}\) \(P(3)\text{,}\) … \(P(k)\) is true, just to show that \(P(k+1)\) is true. Previously, we just assumed \(P(k)\) for this purpose.
in-context