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Old 2010-07-22, 15:58   #1
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May 2005

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Default When is a proof rigurous 'enough' ?

This is something that I have some time thinking, when can one be sure that a proof one has written is enough rigurous that everybody else (or mostly everyone) considers it valid?

Theorem "A"
Every polynomial P(x)=a_n x^n + a_{n-1}x^{n-1}+...+a_1x^1 + a_0 of a single variable of odd degree has at last one real root
As n is odd, if a_n > 0 clearly
\lim_{x\to+\infty}P(x) = +\infty
\lim_{x\to-\infty}P(x) = -\infty
because for |x|>1 the term with the n exponential grows 'faster' than those of less degree. (1)
If a_n < 0 it justs swaps +\infty with -\infty in the limits above.
And because P(x) is continuous (2) for all x (for been a polynomial), then it has to cross the x-axis at last once (3), so a real root exists.

Of course, points (1), (2), and (3) could be developed further (they asume other proofs that I ommited, call them proof "B", "C", and "D"), but should one add those proofs in order for theorem "A" be complete? Could I only give bibliography where those proofs can be found?
In general, do you think I really proved Theorem "A" above? (I think most will say no)


Last fiddled with by Damian on 2010-07-22 at 16:10
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