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The principle of induction states that:

[tex]Suppose\ that\ P(n)\ is\ a\ statement\ involving\ a\ general\ positive\ integer\ n.\ Then\ P(n)\ is\ true\ for\ all\ positive\ integers\ 1,2,3,...\ if\ [/tex]

[tex] (i)\ P(1)\ is\ true,\ and [/tex]

[tex] (ii)\ P(k) \rightarrow P(k+1)\ for\ all\ positive\ integers\ k. [/tex]

I found it a bit circular to prove that the statement P(n) holds true for all n that is an element of natural numbers by presupposing P(k) is true for all k that is an element natural numbers.

That is what I thought at first, but I'm thinking that that's false, since what the principle of induction states in (ii) is that the

I'm still not too comfortable with the principle of induction, could anyone take a look on my thoughts?

[tex]Suppose\ that\ P(n)\ is\ a\ statement\ involving\ a\ general\ positive\ integer\ n.\ Then\ P(n)\ is\ true\ for\ all\ positive\ integers\ 1,2,3,...\ if\ [/tex]

[tex] (i)\ P(1)\ is\ true,\ and [/tex]

[tex] (ii)\ P(k) \rightarrow P(k+1)\ for\ all\ positive\ integers\ k. [/tex]

I found it a bit circular to prove that the statement P(n) holds true for all n that is an element of natural numbers by presupposing P(k) is true for all k that is an element natural numbers.

That is what I thought at first, but I'm thinking that that's false, since what the principle of induction states in (ii) is that the

*implication*itself is what needs to be true.I'm still not too comfortable with the principle of induction, could anyone take a look on my thoughts?

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