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=== Technique of Induction ===

=== Technique of Induction ===

Both Weak and Strong Induction are based on the [[Peano's Axioms|Peano Axioms]], especially the ''Induction Axiom'':<blockquote>If a set S of numbers contains zero and the successor of every number in S, then S contains every number. This is known as the induction axiom. </blockquote>

Both Weak and Strong Induction are based on the [[Peano's Axioms|Peano Axioms]], especially the ''Induction Axiom'':<blockquote>If a set S of numbers contains zero and the successor of every number in S, then S contains every number. </blockquote>

If you want to show that an hypothesis A is true for all [[Natural Numbers]], you look at the set S of all numbers for which the hypothesis is true: you show, that <ol><li>zero is an element of the set S <br>

If you want to show that an hypothesis A is true for all [[Natural Numbers]], you look at the set S of all numbers for which the hypothesis is true: you show, that <ol><li>zero is an element of the set S <br>

<math>0 \in S</math><li>with any number n, S contains the successor of n <br><math> n \in S \Rightarrow (n+1) \in S </math></ol>  

<math>0 \in S</math><li>with any number n, S contains the successor of n <br><math> n \in S \Rightarrow (n+1) \in S </math></ol>  

<math>\Leftrightarrow</math><br>

<math>\Leftrightarrow</math><br>

<math>1 + 2 + 3 + ... + n + (n+1)=\frac{n(n+1)}{2} + \frac{2(n+1)}{2}=\frac{n(n+1)+2(n+1)}{2}=\frac{(n+1)(n+2)}{2}</math><p>

<math>1 + 2 + 3 + ... + n + (n+1)=\frac{n(n+1)}{2} + \frac{2(n+1)}{2}=\frac{n(n+1)+2(n+1)}{2}=\frac{(n+1)(n+2)}{2}</math><p>

Now, we're finished: the hypothesis A hold for all the Natural Numbers.

Now, we're finished: the hypothesis A holds for all the Natural Numbers.