Is it valid to make this stronger assumption? Remember, in induction we are attempting to prove that \(P(n)\) is true for all \(n\text{.}\) What if that were not the case? Then there would be some first \(n_0\) for which \(P(n_0)\) was false. Since \(n_0\) is the first counterexample, we know that \(P(n)\) is true for all \(n \lt n_0\text{.}\) Now we proceed to prove that \(P(n_0)\) is actually true, based on the assumption that \(P(n)\) is true for all smaller \(n\text{.}\)
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