0
$\begingroup$

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 $x$.

$\endgroup$
  • 1
    $\begingroup$ $$\frac{x}{2} < \frac{x}{2} + \frac{x}{2}.$$ $\endgroup$ – Cameron Williams Oct 24 '15 at 4:59
  • $\begingroup$ How do we know this is true? $\endgroup$ – Sorfosh Oct 24 '15 at 5:00
  • 1
    $\begingroup$ You may want to look at the Peano axioms $\endgroup$ – Samrat Mukhopadhyay Oct 24 '15 at 5:01
  • $\begingroup$ You would have to prove that if $c > 0$ and $a < b$, then $a+c < b+c$. $\endgroup$ – Cameron Williams Oct 24 '15 at 5:01
  • $\begingroup$ So it is an axiom? That's great. Thanks $\endgroup$ – Sorfosh Oct 24 '15 at 5:02
2
$\begingroup$

$$ \begin{aligned} x \phantom{\:+0} &= x \\ 0 &< \phantom{x+\:} x \\ x + 0&< x + x \\ x/2 &< x/2 + x/2 \\ x/2 &< x \end{aligned} $$

$\endgroup$
  • $\begingroup$ Oh god, this was easy. Thanks a bunch :D $\endgroup$ – Sorfosh Oct 24 '15 at 5:10
  • $\begingroup$ 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! $\endgroup$ – eyqs Oct 24 '15 at 5:11
  • $\begingroup$ It's not homework, just something i thought of. $\endgroup$ – Sorfosh Oct 24 '15 at 5:12
  • $\begingroup$ 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! $\endgroup$ – eyqs Oct 24 '15 at 5:13
  • $\begingroup$ These are easy, i don't know why i did not think of that. Brian fart i guess $\endgroup$ – Sorfosh Oct 24 '15 at 5:14
1
$\begingroup$

Note that if:

$$x>0$$

Then add $x$ to both sides:

$$x+x>0+x$$

Or

$$2x>x$$

Then we may divide by $2$

$$x>\frac{x}{2}$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.