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Damian · 2010-07-22, 15:58

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?

Example:
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
Proof
As $n$ is odd, if $a_n > 0$ clearly
$\lim_{x\to+\infty}P(x) = +\infty$
and
$\lim_{x\to-\infty}P(x) = -\infty$
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.
$\box$

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)

Thanks,
Damián.

2010-07-22, 15:58	#1
Damian May 2005 Argentina 2×3×31 Posts	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? Example: 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 Proof As $n$ is odd, if $a_n > 0$ clearly $\lim_{x\to+\infty}P(x) = +\infty$ and $\lim_{x\to-\infty}P(x) = -\infty$ 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. $\box$ 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) Thanks, Damián. Last fiddled with by Damian on 2010-07-22 at 16:10