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``mcs'' --2015/5/18 --1:43 -- page 12 --\# 20
{\it 12 Chapter 1 What is a Proo}.7
Example
Theorem 1.5.1. {\it If} $0\underline{<}x\underline{<}2, then-x^{3}+4x+1>0.$
Before we write a proof of this theorem, we have to do some scratchwork to figure out why it is true.
The inequality certainly holds for $x=0$; then the left side is equal to 1 and $1>0$. As $x$ grows, the $4x$ term (which is positive) initially seems to have greater magnitude than $-x^{3}$ (which is negative). For example, when $x=1$, we have $4x=4$, but $-x^{3}=-1$ only. In fact, it looks like $-x^{3}$ doesn't begin to dominate until $x>2$. So it seems the $-x^{3}+4x$ part should be nonnegative for all $x$ between $0$ and 2, which would imply that $-x^{3}+4x+1$ is positive.
So far, so good. But we still have to replace all those ``seems like'' phrases with solid, logical arguments. We can get a better handle on the critical $-x^{3}+4x$ part by factoring it, which is not too hard:
$$
-x^{3}+4x=x(2-x)(2+x)
$$
Aha! For $x$ between $0$ and 2, all of the terms on the right side are nonnegative. And a product of nonnegative terms is also nonnegative. Let's organize this blizzard of observations into a clean proof.
{\it Proo}. Assume $0\underline{<}x\underline{<}2$. Then $x, 2-x$, and $2+x$ are all nonnegative. Therefore, the product of these terms is also nonnegative. Adding 1 to this product gives a positive number, so:
$$
x(2-x)(2+x)+1>0
$$
Multiplying out on the left side proves that
$$
-x^{3}+4x+1>0
$$
as claimed. $\blacksquare$
There are a couple points here that apply to all proofs:
$\bullet$ You'll often need to do some scratchwork while you're trying to figure out the logical steps of a proof. Your scratchwork can be as disorganized as you like--full of dead-ends, strange diagrams, obscene words, whatever. But keep your scratchwork separate from your final proof, which should be clear and concise.
$\bullet$ Proofs typically begin with the word ``Proof'' and end with some sort of de- limiter like $\square $ or ``QED.'' The only purpose for these conventions is to clarify where proofs begin and end.
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