Elementary proof on vector algebra

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Could you please help me verify my proof for this very basic result in vector algebra? Are these steps, mathematically correct or is there a better way to prove this result?

Using the triangle inequality for vectors -

$$|u+v| le |u| + |v|$$

prove that :

$$|a-b| ge ||a|-|b||$$

Proof.

We have $|u+v|le|u|+|v|$.

(1) Let $u=a-b$ and $v=b$. Then,

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

$|a-b| ge |a| - |b|$

(2) Next, let $v = b-a$ and $u = a$. Then,

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

$|b| - |a| le |a-b|$

$|a-b| ge |b| - |a| = -(|a|-|b|)$

Combining (1) and (2), we must have :

$$|a-b| ge ||a| - |b||$$

proof-verification inequality vectors

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edited Jul 25 at 18:13

José Carlos Santos

113k1696174

asked Jul 25 at 18:08

Quasar

697412

add a comment | 

up vote
1
down vote

favorite

Could you please help me verify my proof for this very basic result in vector algebra? Are these steps, mathematically correct or is there a better way to prove this result?

Using the triangle inequality for vectors -

$$|u+v| le |u| + |v|$$

prove that :

$$|a-b| ge ||a|-|b||$$

Proof.

We have $|u+v|le|u|+|v|$.

(1) Let $u=a-b$ and $v=b$. Then,

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

$|a-b| ge |a| - |b|$

(2) Next, let $v = b-a$ and $u = a$. Then,

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

$|b| - |a| le |a-b|$

$|a-b| ge |b| - |a| = -(|a|-|b|)$

Combining (1) and (2), we must have :

$$|a-b| ge ||a| - |b||$$

proof-verification inequality vectors

share|cite|improve this question

edited Jul 25 at 18:13

José Carlos Santos

113k1696174

asked Jul 25 at 18:08

Quasar

697412

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Could you please help me verify my proof for this very basic result in vector algebra? Are these steps, mathematically correct or is there a better way to prove this result?

Using the triangle inequality for vectors -

$$|u+v| le |u| + |v|$$

prove that :

$$|a-b| ge ||a|-|b||$$

Proof.

We have $|u+v|le|u|+|v|$.

(1) Let $u=a-b$ and $v=b$. Then,

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

$|a-b| ge |a| - |b|$

(2) Next, let $v = b-a$ and $u = a$. Then,

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

$|b| - |a| le |a-b|$

$|a-b| ge |b| - |a| = -(|a|-|b|)$

Combining (1) and (2), we must have :

$$|a-b| ge ||a| - |b||$$

proof-verification inequality vectors

share|cite|improve this question

edited Jul 25 at 18:13

José Carlos Santos

113k1696174

asked Jul 25 at 18:08

Quasar

697412

Could you please help me verify my proof for this very basic result in vector algebra? Are these steps, mathematically correct or is there a better way to prove this result?

Using the triangle inequality for vectors -

$$|u+v| le |u| + |v|$$

prove that :

$$|a-b| ge ||a|-|b||$$

Proof.

We have $|u+v|le|u|+|v|$.

(1) Let $u=a-b$ and $v=b$. Then,

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

$|a-b| ge |a| - |b|$

(2) Next, let $v = b-a$ and $u = a$. Then,

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

$|b| - |a| le |a-b|$

$|a-b| ge |b| - |a| = -(|a|-|b|)$

Combining (1) and (2), we must have :

$$|a-b| ge ||a| - |b||$$

proof-verification inequality vectors

share|cite|improve this question

edited Jul 25 at 18:13

José Carlos Santos

113k1696174

asked Jul 25 at 18:08

Quasar

697412

share|cite|improve this question

share|cite|improve this question

edited Jul 25 at 18:13

José Carlos Santos

113k1696174

edited Jul 25 at 18:13

José Carlos Santos

113k1696174

edited Jul 25 at 18:13

José Carlos Santos

113k1696174

113k1696174

asked Jul 25 at 18:08

Quasar

697412

asked Jul 25 at 18:08

Quasar

697412

asked Jul 25 at 18:08

Quasar

697412

697412

add a comment | 

add a comment | 

1 Answer
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accepted

The proof is correct, but you should get used to put $implies$ and $iff$ whenever it is need. For instance: after putting $u=a-b$ and $v=b$, it would have been better if you had written that $|a|leqslant|a-b|+|b|$ and that$$|a|leqslant|a-b|+|b|iff|a-b|geqslant|a|-|b|.$$

share|cite|improve this answer

answered Jul 25 at 18:11

José Carlos Santos

113k1696174

add a comment | 

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
3
down vote



accepted

The proof is correct, but you should get used to put $implies$ and $iff$ whenever it is need. For instance: after putting $u=a-b$ and $v=b$, it would have been better if you had written that $|a|leqslant|a-b|+|b|$ and that$$|a|leqslant|a-b|+|b|iff|a-b|geqslant|a|-|b|.$$

share|cite|improve this answer

answered Jul 25 at 18:11

José Carlos Santos

113k1696174

add a comment | 

up vote
3
down vote



accepted

The proof is correct, but you should get used to put $implies$ and $iff$ whenever it is need. For instance: after putting $u=a-b$ and $v=b$, it would have been better if you had written that $|a|leqslant|a-b|+|b|$ and that$$|a|leqslant|a-b|+|b|iff|a-b|geqslant|a|-|b|.$$

share|cite|improve this answer

answered Jul 25 at 18:11

José Carlos Santos

113k1696174

add a comment | 

up vote
3
down vote



accepted

up vote
3
down vote



accepted

The proof is correct, but you should get used to put $implies$ and $iff$ whenever it is need. For instance: after putting $u=a-b$ and $v=b$, it would have been better if you had written that $|a|leqslant|a-b|+|b|$ and that$$|a|leqslant|a-b|+|b|iff|a-b|geqslant|a|-|b|.$$

share|cite|improve this answer

answered Jul 25 at 18:11

José Carlos Santos

113k1696174

The proof is correct, but you should get used to put $implies$ and $iff$ whenever it is need. For instance: after putting $u=a-b$ and $v=b$, it would have been better if you had written that $|a|leqslant|a-b|+|b|$ and that$$|a|leqslant|a-b|+|b|iff|a-b|geqslant|a|-|b|.$$

share|cite|improve this answer

answered Jul 25 at 18:11

José Carlos Santos

113k1696174

share|cite|improve this answer

share|cite|improve this answer

answered Jul 25 at 18:11

José Carlos Santos

113k1696174

answered Jul 25 at 18:11

José Carlos Santos

113k1696174

answered Jul 25 at 18:11

José Carlos Santos

113k1696174

113k1696174

add a comment | 

add a comment | 

 

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