Mixing
When do absolute values $| cdot|$ retain inequalities? What about for norms?
$begingroup$
Can I substitute stuff inside $| cdot|$ and retain inequalities?
Example: want to make
$$
left|frac{m}{n} - 1 right|
< epsilon,
quad epsilon > 0; quad n,m in mathbb{N}$$
Could I say:
Take
$$
n > frac{m}{epsilon+1}
$$
so that
$$
left|frac{m}{n} - 1 right|
< left| frac{m}{frac{m}{epsilon+1}} - 1 right|
= |epsilon + 1 - 1|
= epsilon.
$$
Now for this simple case this seems to work, since for a fixed and positive constant $1$ it's safely known that the distance of a larger positive quantity from that will lead to larger distance.
However, I don't think the $| cdot |$ would always allow this kind of "positive real line" modifications?
Any background to this "property" or non-property of $|cdot|$ or why not norms as well? What would one test for? That the operation is such that it's explainable either by homogeneity or triangle eq. (or possible C-S)?
In general, when do absolute values and norms retain inequalities?
absolute-value
asked Jan 19 at 16:35
mavaviljmavavilj
2,81911137
$endgroup$
add a comment |
$begingroup$
Can I substitute stuff inside $| cdot|$ and retain inequalities?
Example: want to make
$$
left|frac{m}{n} - 1 right|
< epsilon,
quad epsilon > 0; quad n,m in mathbb{N}$$
Could I say:
Take
$$
n > frac{m}{epsilon+1}
$$
so that
$$
left|frac{m}{n} - 1 right|
< left| frac{m}{frac{m}{epsilon+1}} - 1 right|
= |epsilon + 1 - 1|
= epsilon.
$$
Now for this simple case this seems to work, since for a fixed and positive constant $1$ it's safely known that the distance of a larger positive quantity from that will lead to larger distance.
However, I don't think the $| cdot |$ would always allow this kind of "positive real line" modifications?
Any background to this "property" or non-property of $|cdot|$ or why not norms as well? What would one test for? That the operation is such that it's explainable either by homogeneity or triangle eq. (or possible C-S)?
In general, when do absolute values and norms retain inequalities?
absolute-value
asked Jan 19 at 16:35
mavaviljmavavilj
2,81911137
$endgroup$
3
$begingroup$
Yes, $|x|<1$ implies $x<1$. No, $x<1$ does not imply $|x|<1$. Yes, $|x|<1$ and $x$ nonnegative imply $x<1$. This is all there is to it...
$endgroup$
– Did
Jan 19 at 16:37
1
$begingroup$
$0 < 1$ but $|0-2| > |1-2|$
$endgroup$
– mathworker21
Jan 19 at 16:37
add a comment |
$begingroup$
Can I substitute stuff inside $| cdot|$ and retain inequalities?
Example: want to make
$$
left|frac{m}{n} - 1 right|
< epsilon,
quad epsilon > 0; quad n,m in mathbb{N}$$
Could I say:
Take
$$
n > frac{m}{epsilon+1}
$$
so that
$$
left|frac{m}{n} - 1 right|
< left| frac{m}{frac{m}{epsilon+1}} - 1 right|
= |epsilon + 1 - 1|
= epsilon.
$$
Now for this simple case this seems to work, since for a fixed and positive constant $1$ it's safely known that the distance of a larger positive quantity from that will lead to larger distance.
However, I don't think the $| cdot |$ would always allow this kind of "positive real line" modifications?
Any background to this "property" or non-property of $|cdot|$ or why not norms as well? What would one test for? That the operation is such that it's explainable either by homogeneity or triangle eq. (or possible C-S)?
In general, when do absolute values and norms retain inequalities?
absolute-value
asked Jan 19 at 16:35
mavaviljmavavilj
2,81911137
$endgroup$
Can I substitute stuff inside $| cdot|$ and retain inequalities?
Example: want to make
$$
left|frac{m}{n} - 1 right|
< epsilon,
quad epsilon > 0; quad n,m in mathbb{N}$$
Could I say:
Take
$$
n > frac{m}{epsilon+1}
$$
so that
$$
left|frac{m}{n} - 1 right|
< left| frac{m}{frac{m}{epsilon+1}} - 1 right|
= |epsilon + 1 - 1|
= epsilon.
$$
Now for this simple case this seems to work, since for a fixed and positive constant $1$ it's safely known that the distance of a larger positive quantity from that will lead to larger distance.
However, I don't think the $| cdot |$ would always allow this kind of "positive real line" modifications?
Any background to this "property" or non-property of $|cdot|$ or why not norms as well? What would one test for? That the operation is such that it's explainable either by homogeneity or triangle eq. (or possible C-S)?
In general, when do absolute values and norms retain inequalities?
absolute-value
absolute-value
asked Jan 19 at 16:35
mavaviljmavavilj
2,81911137
asked Jan 19 at 16:35
mavaviljmavavilj
2,81911137
edited Jan 21 at 11:21
mavavilj
asked Jan 19 at 16:35
mavaviljmavavilj
2,81911137
asked Jan 19 at 16:35
mavaviljmavavilj
2,81911137
asked Jan 19 at 16:35
mavaviljmavavilj
2,81911137
2,81911137
3
$begingroup$
Yes, $|x|<1$ implies $x<1$. No, $x<1$ does not imply $|x|<1$. Yes, $|x|<1$ and $x$ nonnegative imply $x<1$. This is all there is to it...
$endgroup$
– Did
Jan 19 at 16:37
1
$begingroup$
$0 < 1$ but $|0-2| > |1-2|$
$endgroup$
– mathworker21
Jan 19 at 16:37
add a comment |
3
$begingroup$
Yes, $|x|<1$ implies $x<1$. No, $x<1$ does not imply $|x|<1$. Yes, $|x|<1$ and $x$ nonnegative imply $x<1$. This is all there is to it...
$endgroup$
– Did
Jan 19 at 16:37
1
$begingroup$
$0 < 1$ but $|0-2| > |1-2|$
$endgroup$
– mathworker21
Jan 19 at 16:37
3
3
$begingroup$
Yes, $|x|<1$ implies $x<1$. No, $x<1$ does not imply $|x|<1$. Yes, $|x|<1$ and $x$ nonnegative imply $x<1$. This is all there is to it...
$endgroup$
– Did
Jan 19 at 16:37
$begingroup$
Yes, $|x|<1$ implies $x<1$. No, $x<1$ does not imply $|x|<1$. Yes, $|x|<1$ and $x$ nonnegative imply $x<1$. This is all there is to it...
$endgroup$
– Did
Jan 19 at 16:37
1
1
$begingroup$
$0 < 1$ but $|0-2| > |1-2|$
$endgroup$
– mathworker21
Jan 19 at 16:37
$begingroup$
$0 < 1$ but $|0-2| > |1-2|$
$endgroup$
– mathworker21
Jan 19 at 16:37
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3079539%2fwhen-do-absolute-values-cdot-retain-inequalities-what-about-for-norms%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3079539%2fwhen-do-absolute-values-cdot-retain-inequalities-what-about-for-norms%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
3
$begingroup$
Yes, $|x|<1$ implies $x<1$. No, $x<1$ does not imply $|x|<1$. Yes, $|x|<1$ and $x$ nonnegative imply $x<1$. This is all there is to it...
$endgroup$
– Did
Jan 19 at 16:37
1
$begingroup$
$0 < 1$ but $|0-2| > |1-2|$
$endgroup$
– mathworker21
Jan 19 at 16:37