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I am trying to understand this proof for using induction. Please help me!!

As per the book "Alan F beardon, Abstract algebra and geometry" The following....

Quote:

Proof: Let B be the set of positive integers that are not in A. Suppose that

B = ∅; then, by the Well-Ordering Principle, B has a smallest element, say b.

As before, b ≥ 2, so that now {1, . . . , b − 1} ⊂ A. With the new hypothesis,

this implies that b ∈ A which is again a contradiction. Thus (as before) B = ∅,and A = N.

Questions??

b is >= 2 because 1 is in A right?

b - 1 is 1 right?? therefore it should be in A??

Then....

b - 1 is an element of A so b is an element of A + 1??

so how does b become an element of A??

Danke....

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# Simple Proof for using Induction

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