Induction is really an axiom, and like all axioms, it is self evident. Lets break it down.
- let P(n) be a predicate
- if P(0) is true and for all natural numbers n, (P(n)=>P(n+1)) is true
- then for all natural numbers n, P(n) is true!
So if P(0) is true, and P(0) => P(1) is true, then P(1)=> P(2) is true, and P(2) => P(3) is true... and so on...
Now lets prove the following proposition:
It's interesting to note, that I didn't really introduce anything new here... We were already given the equation sum of i = n(n+1)/2, and we simply proved that it's true. However, induction did not give us that equation nor did it give us the reasoning behind it. Is that satisfying?
Now lets prove the following proposition:
It's interesting to note, that I didn't really introduce anything new here... We were already given the equation sum of i = n(n+1)/2, and we simply proved that it's true. However, induction did not give us that equation nor did it give us the reasoning behind it. Is that satisfying?