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You might be interested in reading about transfinite induction, which is an interesting and legitimate mathematical proof technique. However your 'induction' do
by tgb 10y ago
You might be interested in reading about transfinite induction, which is an interesting and legitimate mathematical proof technique. However your 'induction' does not succeed in doing this. Transfinite induction requires two induction steps, you've only provided one of them. Not that you would actually use transfinite induction for anything resembling this, that's what calculus and analysis is for.
For example your strategy would show that the sequence of positive integers 1, 2, 3... has a finite limit! After all, each one is finite so by induction the limit is finite too. Except of course no one would expect this induction to apply to the limit as well as to the individual steps.
- kazinator 10y agoI'm not surprised that something called "induction" doesn't succeed in achieving what "transfinite induction" can do, otherwise the latter couldn't exist as a separate technique with its own name. :) > For example your strategy would show that the sequence of positive integers 1, 2, 3... has a finite limit! How so? Induction is in fact the basis for the common proof that there is no highest integer: for any integer k, we can add 1 to find a larger integer. Every integer k has the property P(k) := "k is not the highest integer".