Could someone provide me a valid proof that $\frac{x}{2}$ is smaller than $x$. It seems obvious but i cannot think of a proof. Or just prove that $x+x$ is larger than $x$ for positive $x$.
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1$$\frac{x}{2} < \frac{x}{2} + \frac{x}{2}.$$ – Cameron Williams Oct 24 '15 at 04:59
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How do we know this is true? – Sorfosh Oct 24 '15 at 05:00
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1You may want to look at the Peano axioms – Samrat Mukhopadhyay Oct 24 '15 at 05:01
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You would have to prove that if $c > 0$ and $a < b$, then $a+c < b+c$. – Cameron Williams Oct 24 '15 at 05:01
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So it is an axiom? That's great. Thanks – Sorfosh Oct 24 '15 at 05:02
2 Answers
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$$ \begin{aligned} x \phantom{\:+0} &= x \\ 0 &< \phantom{x+\:} x \\ x + 0&< x + x \\ x/2 &< x/2 + x/2 \\ x/2 &< x \end{aligned} $$
eyqs
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You're welcome. You could try just writing down Cameron's comment if this is homework or something and see what your teacher gives you. I doubt they'll require you to write down something as unnecessary as mine! – eyqs Oct 24 '15 at 05:11
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Alright. See if you can prove that $x/2 > x$ when $x < 0$, $x^2 < x$ when $|x| < 1$, $\sin x < x$ when $|x| < 1$, and even more! – eyqs Oct 24 '15 at 05:13
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These are easy, i don't know why i did not think of that. Brian fart i guess – Sorfosh Oct 24 '15 at 05:14
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Note that if:
$$x>0$$
Then add $x$ to both sides:
$$x+x>0+x$$
Or
$$2x>x$$
Then we may divide by $2$
$$x>\frac{x}{2}$$