# Prove using a Triangle Inequality

I need to prove $\vert2n^5 − n^3 + 2000|\geq 2n^5 − |n^3 − 2000|,\;$ with $n$ being a natural number.

I understand that I need to use a triangle inequality:

$$|x-y|\geq |x|-|y|$$ but I don't understand how to take them from just using x and y to actually having $x, y$ representing functions of $n$.

Mostly, I don't quite understand how to figure out what $x$ and $y$ are equal to in order to start this proof.

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– mlc
Apr 20 '17 at 22:36
• can you define $n$? Apr 20 '17 at 22:36
• n is a natural number @mrnovice Apr 20 '17 at 22:37
• Let $a = 2n^5$ and $b = n^3 - 2000$ then you are being asked to prove $|a-b| \ge a + |b|$. Can you do that? Apr 20 '17 at 22:41
• Nice edit, @user438836. Mind if I help polish it off with some formatting of the math? I'll try, and you can let me know what you think of it. Apr 20 '17 at 23:01

$$|2n^5-n^3+2000|\geq 2n^5-|n^3-2000|\iff|-2000+n^3|+|2n^5-n^3+2000|\geq2n^5$$

The triangle inequality tells us $$|x|+|y|\geq|x+y|$$

So let $x = -2000+n^3,\quad y=2n^5-n^3+2000,\quad$then:

$x+y = 2n^5,\quad$ and:

$$|x|+|y| = |-2000+n^3|+|2n^5-n^3+2000|\geq|2n^5|=2n^5\quad\text{since n}\in\mathbb{N}$$

You need to use that $$|x-y|\geq x-y \geq x-|y|$$

and now just plug-in: $x=2n^5,$ $y=n^3-2000.$

$|a + b| \le |a| + |b|$

$|a+ b|-|b| \le |a|$

$\pm(a+b) - |b| \le |a|$

$(a+b) - |b| \le |a|$.

Let $a = 2n^5 -n^3 +2000$ and $b = n^3 - 2000$.

Done.

• Ever wonder why your post has not received an up vote even if it is clearly true? Apr 21 '17 at 0:40
• @DeepSea Is that a rhetorical question? Apr 21 '17 at 1:53
• @mrnovice: Sure is. Also, it might be true that the community expects users with high enough reputation points posting some new insights or new answers to even an old or close to an old question to receive an up vote. Apr 21 '17 at 1:57
• No. I can't say that ever even crossed my mind. Apr 21 '17 at 2:22
• If I were to justify what insight my mundane answer has, it's that no insight is actually required or even desirable. This is a straight application with no thought, insight, or new answer required. Apr 21 '17 at 2:30