Deriving Triangle Inequality: Formal Definition of Absolute Value Method

[Taylor_1989]

Homework Statement

Hi guys, I would just like someone to go over my method for this derivation/proof ( not sure of the right word to use here). Anyway I think this is right method, but just feel like I am missing something. Could someone please check my method. Thanks in advance.

Homework Equations

$$|x|= x\geq 0 , -x < 0 $$

$$|a-b|\leq|a|-|b|$$

The Attempt at a Solution

By using the formal definition of the absolute value I get this:
[/B]
1.$$-|a|\leq a\leq |a|$$
2.$$-|b|\leq b\leq |b|$$

1-2: $$-(|a|-|b|)\leq a-b \leq |a|-|b| $$

Therefore I get: $$|a-b|\leq|a|-|b|$$

Is this correct? Is there any improvements that anyone could share. I do have a couple more variations of the triangle inequality to go through but want to try the first before posting.

 

[Ray Vickson]

Homework Statement

Hi guys, I would just like someone to go over my method for this derivation/proof ( not sure of the right word to use here). Anyway I think this is right method, but just feel like I am missing something. Could someone please check my method. Thanks in advance.

Homework Equations

$$|x|= x\geq 0 , -x < 0 $$

$$|a-b|\leq|a|-|b|$$

This is incorrect. Look at the case $a=1, b=2$, or better still, the case $a=1, b=-2$.

 

[mfb]

The first line under "relevant equations" looks odd and the second one is wrong.

You cannot subtract inequalities like that. As an example, 4<5 and 3<7, but 4-3 < 5-7 is wrong.

 

[Austin Hook]

I wish the section for the problem statement were filled out explicitly.

 

[Buffu]

@mfb and @Ray Vickson have already pointed out your mistakes. I just want to add a bit to mfb's reply.

Inequalities with same "symbol" can be added and inequalities with "different symbol" can be subtracted from one another and not the other way around.
Like $13 < 42$ and $-42 < -1$ can be added to produce $-29 < 41$ and $42 > 13$ and $-42 < -1$ can be subtracted to produce $84 > 14$.
When subtracting the symbol of inequality from which the other is subtracted will be the symbol of the resultant inequality.

 

[mfb]

You can make an addition out of the subtraction:

$-42 < -1$ is equivalent to (edit: fixed) $42 > 1$ (reverse the sign on both sides, reverse the direction of the inequality), and that can be added to $42>13$ to $84>14$.

And if you don't like the first step, split it in substeps:
$a<b$
subtract a on both sides
$0 < b-a$
subtract b on both sides
$-b < -a$
Now write it in the other direction:
$-a > -b$.

 

[Taylor_1989]

First thanks for the response and I see the error I made. I have also notice that I put the inequity the wrong way round sorry. So I have had another attempt at the solution which is below:

Prove: $$|a|-|b|\leq|a-b|$$

Attempt:$$a=a-b+b$$
$$|a|=|a-b+b|$$

Using triangle inequity

$$|a|\leq|a-b|+|b|$$
$$|a|-|b|\leq|a-b|$$

 

[haruspex]

First thanks for the response and I see the error I made. I have also notice that I put the inequity the wrong way round sorry. So I have had another attempt at the solution which is below:

Prove: $$|a|-|b|\leq|a-b|$$

Attempt:$$a=a-b+b$$
$$|a|=|a-b+b|$$

Using triangle inequity

$$|a|\leq|a-b|+|b|$$
$$|a|-|b|\leq|a-b|$$

Looks good.