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Subset of a finite set is finite

I see this proof is presented using induction here http://www.proofwiki.org/wiki/Subset_of_Finite_Set_is_Finite

Why do we need to use induction and such a long proof, instead could not we use basic proof by contradiction and let there exist an infinite subset. Then since a set contains its subset it will be infinite too leading to a contradiction.

I am trying to learn proofs and analysis by selfstudy and I don't have professors to help me so can anyone explain me why this proof approach is preferred and is my proof wrong ?

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cesim
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    What's your definition of finite set? What's an infinite set? The proof will depend on the details. – Zhen Lin Jan 14 '13 at 09:54
  • I would have thought it was "obvious" that a subset of a finite set is finite. How could it have more elements than the original set? – TheMathemagician Jan 14 '13 at 10:13
  • @TheMathemagician define "more elements". – John Dvorak Jan 14 '13 at 10:14
  • Isn't the number of elements well-defined for a finite set? – TheMathemagician Jan 14 '13 at 10:15
  • @TheMathemagician then you need to include the definition, and prove it from that definition. – John Dvorak Jan 14 '13 at 10:16
  • The cardinality of a finite set is the unique $n$ such that there exists a bijection between that set and the set of integers $1..N$. You could move from there. A set is finite iff such an $n$ exists. – John Dvorak Jan 14 '13 at 10:19
  • http://math.stackexchange.com/questions/239566/subset-of-a-finite-set-is-finite – Asaf Karagila Jan 14 '13 at 10:23
  • @cesim: The last sentence in your proposed shorter proof presupposes that, whenever a set $S$ has an infinite subset, then $S$ itself is infinite. Unfortunately, that's just a restatement of what you're trying to prove. – Andreas Blass Jan 14 '13 at 15:06
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    Why the votes to reopen? – Chris Eagle Jan 14 '13 at 15:25
  • @ChrisEagle: I vote to reopen. The duplicate question gives two proofs that are based on induction. The OP in the question wants to know "Why do we need to use induction and such a long proof, instead could not we use basic proof by contradiction and let there exist an infinite subset[?]" – Thomas Jan 14 '13 at 16:07

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