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I'm having difficulties with writing proofs, probably because I've just started the subject. And i really would like to avoid looking at the answers and solve it as best as I can myself.

Now I'm asked to give a proof of: $-v \leq u \leq v$ iif $|u| \leq v$

Where I'm currently at is the following reasoning:

$-v \leq u \leq v$ can be rewritten as $0 \leq u + v \leq 2v$. Since we've said that $|u| \leq v$, this is trivially true: since $|u|$ always lesser than or equals to v, it's always lesser than or equals to 2v.

But I'm not quite if I'm actually proving anything with this statement..

Some pointers in the rifght direction would be very much appreciated...

Apeiron
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    The best way is to check the definition of $|u|$. – user64066 Aug 14 '13 at 18:48
  • |u| is defined as u when u >= 0, -u otherwise – Apeiron Aug 14 '13 at 18:50
  • Now try to use your equation with this. Hint: You need to check both cases but you already have $-v\leq u\leq v$ what you need is something related with$-u$. – user64066 Aug 14 '13 at 18:54
  • Ah I think I see it yes, so, basically the way I was heading was kind of fruitless I guess? But now this leaves me with how'd you figuer (apart from experience) to use that approach? – Apeiron Aug 14 '13 at 19:06

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By definition:

$$|u|=\begin{cases}\;\;\,u&,\;\;\text{if}\;\;u\ge 0\\{}\\-u&,\;\;\text{if}\;\;u<0\end{cases}$$

Thus, for $\,v\ge 0\;$ :

$$|u|\le v\iff\begin{cases}u\le v&,\;\; u\ge 0\iff 0\le u\le v\\{}\\-u\le v&,\;\;u<0\iff -v\le u<0\end{cases}$$

From the above we get $\;|u|\le v\iff -v\le u\le v\;$

DonAntonio
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  • What triggered you to choose this method? I mean how do I know where to start (with any proof)? There's a follow up question to this: deliver a proof of the triangle equality using he result. Starts by stating $|x| \leq |x|$, working to $-|x| \leq x \leq |x|$ and adding $-|y| \leq y \leq |y|$ to it et cetera . I would've never thought of this in my life time probably... How do I know where to start? – Apeiron Aug 15 '13 at 08:27
  • The most basic and elementary assumption is to go to definitions. Not always is that necessary, but in this basic case it is enough...and enlighting. Where to start a proof is not always clear/easy/straightforward, but with experience some things seem to be more likely to help than others... – DonAntonio Aug 15 '13 at 13:31
  • Basically: practice, make lots of mistakes and don't get put down when I'm lost :) – Apeiron Aug 15 '13 at 18:05