How can we rigorously prove the proposition "Suppose if in case 1 is true, the equation 4.23 is true"?
For the constants b and j, the implication in green is logical. If the upper limit of j has been set, the equation 4.23 follows directly. However, as n increases, the upper limit of j also increases, but is slower. This is where I find it hard to prove that there is always a value m> 0 such that for all n> = m, equation 4.23 is true.
The problem is in this other question.
Why does it always work? I do not see how we would use induction.
For $ n = $ 3, a quick calculation shows that it works, however, I think that generalizes well.