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Any proof which is proven by direct proof method is easy to understand and feel. How do you get the feel or intuition behind the proof when a theorem or lemma is proved using the principle of mathematical induction?

akr_
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    Are you familiar with the analogy of the row of dominos? Have you searched this site to look for questions about this, such as this one ? – Arturo Magidin Mar 04 '20 at 05:54
  • As an infinite chain of Modus Ponens: (1) from $P(0)$ and $P(0) \to P(1)$ derive $P(1)$; (2) from $P(1)$ and $P(1) \to P(2)$ derive $P(2)$; and so on. The gist of Induction is to formalize "and so on". – Mauro ALLEGRANZA Mar 04 '20 at 07:04
  • @ArturoMagidin No, I haven't. I will have a look at it . Thanks – akr_ Mar 05 '20 at 07:31

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If you are talking intuition:

In a proof by induction we prove two things.

2) If something is true for one value, it will always have to be true for the next value.

1) Something it true for the first value.

.....

To me it is very intuitive that that means something is true for the first value (by step 1), so it is true for the second value (by step 2), so it is true for the third value (by step 2 again), so it is true by the fourth value (by step 2 yet again), and therefore by repeating step 2) any number of times, it is true for any value we can count to.

And intuitively, we can (given enough time) count to any positive whole number (supposedly). So give those two facts, it intuitively follows the something is true for all values.

fleablood
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  • (+1) for "(given enough time)". Certainly intuitive enough while at the same can rankle pure set theoreticians. and the "(supposedly)" really seals the deal. – CopyPasteIt Mar 04 '20 at 16:09
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    Hmmph.... I was going add that the "given enough time" doesn't actually matter because that "time" is only the time it takes for use to count them, not any time for the statement to be true. The statement (if true) is always true whether we take the time to prove it or not. ... But I figured I was going off on nitpick that would confuse the OP.... (I don't really hold much truck with pure set theoreticians...) – fleablood Mar 04 '20 at 16:25
  • I didn't ask why principle of induction is a correct proof method. How do you prove "If x is odd then x^2 is odd."? If you go on to prove this then you get the feeling why the implication is true given the premise (because it is a direct proof method) but it is not the same when you prove something inductively. Anyways thanks for taking your time to answer. I will have to look elsewhere for what I am looking. – akr_ Mar 04 '20 at 18:36
  • You asked for an intuition and a way to "feel" that makes a proof by induction feels right. I believe I provided exactly that. If you know that every rung of a ladder leads to the next one, then it feels utterly correct all rungs of the ladder are true, doesn't it? I can't imagine why that wouldn't feel utterly true. Maybe you should explain what isn't intuitive about a proof by induction. How can something possibly be false if it is true every single step of the way? – fleablood Mar 04 '20 at 19:04