- #1

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- TL;DR Summary
- Induction Principle.

In this video lecture (though I have linked the video at "current time", in case it doesn't; work please see the video at 19:16), the lecturer just works out (he is not explaining anything) the proof of Induction Principle starting from ##N##. Let me give out here what he did:

1. P(N) holds

2. P(n) implies P(n+1)

then P(n) is true for all ##n \gt N##.

I couldn't understand anything of step 2 of the proof. Can you please give some hint?

*Statement: Let N be an integer*, ##P(n)## denotes the Property P that a natural number ##n \geq N## may or may not follow. If1. P(N) holds

2. P(n) implies P(n+1)

then P(n) is true for all ##n \gt N##.

*Proof: ##Q(n) = P(n+N-1)##*

1. ## Q(1) = P(N)##, since P(N) holds therefore, Q(1) is true.

2. If ## Q(n) = P(n+N-1)## holds then ##Q(n+1) = P(n+N)## holds and therefore P(n) holds for all ##n\geq N##.1. ## Q(1) = P(N)##, since P(N) holds therefore, Q(1) is true.

2. If ## Q(n) = P(n+N-1)## holds then ##Q(n+1) = P(n+N)## holds and therefore P(n) holds for all ##n\geq N##.

I couldn't understand anything of step 2 of the proof. Can you please give some hint?