Fdtxjr

Schwarzchild Radius of the Universe

Patience, young "Padovan"

What are these boxed doors outside store fronts in New York?

How did the USSR manage to innovate in an environment characterized by government censorship and high bureaucracy?

Copycat chess is back

Is it possible to make sharp wind that can cut stuff from afar?

Shell script can be run only with sh command

How to type dʒ symbol (IPA) on Mac?

Is it tax fraud for an individual to declare non-taxable revenue as taxable income? (US tax laws)

A function which translates a sentence to title-case

What is the command to reset a PC without deleting any files

Accidentally leaked the solution to an assignment, what to do now? (I'm the prof)

Is there a familial term for apples and pears?

Japan - Plan around max visa duration

Motorized valve interfering with button?

Why doesn't Newton's third law mean a person bounces back to where they started when they hit the ground?

How can I fix this gap between bookcases I made?

What do you call something that goes against the spirit of the law, but is legal when interpreting the law to the letter?

Possibly bubble sort algorithm

Copenhagen passport control - US citizen

Why is an old chain unsafe?

How do you conduct xenoanthropology after first contact?

How long does it take to type this?

How is the claim "I am in New York only if I am in America" the same as "If I am in New York, then I am in America?

To prove that $ cap_n=1^infty (0 , frac1n) = emptyset $

To show that $f(x)= frac1x$ is not uniformly continuous on $(0,infty)$Prove that $sigma$-algebra of subsets of $mathbbR$ of the form $(a,infty)$ contains all the intervals.Prove that $bigcap_n=1^infty$ $(0,frac1 n) = emptyset$Show that there is a unique $r$ in real separates $A$ and $B$, such that $(-infty, r)subset A$ and $(r,infty)subset B$.Prove that $bigcaplimits^infty_n=1 left(0, frac1n right)= emptyset$Disproving the proposition that $a_n = (-1)^nn$ has a limit.To prove $cap_n=1^infty A_n $ is non emptyA sequence of nested unbounded closed intervals $L_1supseteq L_2supseteq L_3supseteqcdots$ with $bigcap_n=1^inftyL_n = varnothing$Let $a_n_n = 1^infty$ and $b_n_n = 1^infty$ be two sequences of real numbers s.t $|a_n -b_n| < frac1n$Using Archimedian property to prove that infimum of set is $0$

4

$begingroup$

$$ bigcap_n=1^infty left(0 , frac1nright) = varnothing$$

Now assume that intersection contains $b$. So, $ b < frac1n forall n in N$. Since $b > 0$, so we have by Archimedian property that $exists n in mathbbN$ such that $ frac1n < b$ which is a contradiction to assumption that $b < frac1n$

Is this correct ?

Thanks

real-analysis self-learning

share|cite|improve this question

edited Mar 22 at 8:46

José Carlos Santos

173k23133241

asked Mar 22 at 8:40

J. DeffJ. Deff

717519

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's right. The same reasoning I used for a different problem.
  $endgroup$
  – Balakrishnan Rajan
  Mar 22 at 8:43
  

  
  
  
  

add a comment | 

4

$begingroup$

$$ bigcap_n=1^infty left(0 , frac1nright) = varnothing$$

Now assume that intersection contains $b$. So, $ b < frac1n forall n in N$. Since $b > 0$, so we have by Archimedian property that $exists n in mathbbN$ such that $ frac1n < b$ which is a contradiction to assumption that $b < frac1n$

Is this correct ?

Thanks

real-analysis self-learning

share|cite|improve this question

edited Mar 22 at 8:46

José Carlos Santos

173k23133241

asked Mar 22 at 8:40

J. DeffJ. Deff

717519

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's right. The same reasoning I used for a different problem.
  $endgroup$
  – Balakrishnan Rajan
  Mar 22 at 8:43
  

  
  
  
  

add a comment | 

$begingroup$

$$ bigcap_n=1^infty left(0 , frac1nright) = varnothing$$

Now assume that intersection contains $b$. So, $ b < frac1n forall n in N$. Since $b > 0$, so we have by Archimedian property that $exists n in mathbbN$ such that $ frac1n < b$ which is a contradiction to assumption that $b < frac1n$

Is this correct ?

Thanks

real-analysis self-learning

share|cite|improve this question

edited Mar 22 at 8:46

José Carlos Santos

173k23133241

asked Mar 22 at 8:40

J. DeffJ. Deff

717519

$endgroup$

share|cite|improve this question

edited Mar 22 at 8:46

José Carlos Santos

173k23133241

asked Mar 22 at 8:40

J. DeffJ. Deff

717519

share|cite|improve this question

edited Mar 22 at 8:46

José Carlos Santos

173k23133241

asked Mar 22 at 8:40

J. DeffJ. Deff

717519

edited Mar 22 at 8:46

José Carlos Santos

173k23133241

edited Mar 22 at 8:46

José Carlos Santos

173k23133241

asked Mar 22 at 8:40

J. DeffJ. Deff

717519

asked Mar 22 at 8:40

J. DeffJ. Deff

717519

1 Answer
1

active

oldest

votes

2

$begingroup$

Looks good to me.

Small notational nitpicks: It's Bbb N ("BlackBoard Bold") to make $Bbb N$.

And when using symbols to convey logic, like in your $b<frac1nforall nin Bbb N$, the quantifiers ($forall$ and $exists$) always come before whatever it is they modify.

So while one in English could say "$b$ is smaller than $frac1n$ for any natural number $n$", the symbolic statement must be $forall nin Bbb N, b<frac1n$ (exactly how to separate $forall nin Bbb N$ and $b<frac1n$ is up to you, you can use a comma like I did, or a colon, or wrap $b<frac1n$ in parentheses).

share|cite|improve this answer

answered Mar 22 at 8:50

ArthurArthur

122k7122211

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  "..thus we have $f$ continuous for every $xin D$.." opposed to "..thus for every $xin D$ $f$ is continuous.." or do you regard worded and symbolised quantifications differently?
  $endgroup$
  – Alvin Lepik
  Mar 22 at 8:53
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @AlvinLepik I definitely regard worded and symbolic quantifiers differently. When using regular, human English, putting quantifiers after is natural, because that's how English semantics work.
  $endgroup$
  – Arthur
  Mar 22 at 8:54
  
  

  
  
  
  

add a comment | 

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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3157899%2fto-prove-that-cap-n-1-infty-0-frac1n-emptyset%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

2

$begingroup$

Looks good to me.

Small notational nitpicks: It's Bbb N ("BlackBoard Bold") to make $Bbb N$.

And when using symbols to convey logic, like in your $b<frac1nforall nin Bbb N$, the quantifiers ($forall$ and $exists$) always come before whatever it is they modify.

So while one in English could say "$b$ is smaller than $frac1n$ for any natural number $n$", the symbolic statement must be $forall nin Bbb N, b<frac1n$ (exactly how to separate $forall nin Bbb N$ and $b<frac1n$ is up to you, you can use a comma like I did, or a colon, or wrap $b<frac1n$ in parentheses).

share|cite|improve this answer

answered Mar 22 at 8:50

ArthurArthur

122k7122211

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  "..thus we have $f$ continuous for every $xin D$.." opposed to "..thus for every $xin D$ $f$ is continuous.." or do you regard worded and symbolised quantifications differently?
  $endgroup$
  – Alvin Lepik
  Mar 22 at 8:53
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @AlvinLepik I definitely regard worded and symbolic quantifiers differently. When using regular, human English, putting quantifiers after is natural, because that's how English semantics work.
  $endgroup$
  – Arthur
  Mar 22 at 8:54
  
  

  
  
  
  

add a comment | 

2

$begingroup$

Looks good to me.

Small notational nitpicks: It's Bbb N ("BlackBoard Bold") to make $Bbb N$.

And when using symbols to convey logic, like in your $b<frac1nforall nin Bbb N$, the quantifiers ($forall$ and $exists$) always come before whatever it is they modify.

So while one in English could say "$b$ is smaller than $frac1n$ for any natural number $n$", the symbolic statement must be $forall nin Bbb N, b<frac1n$ (exactly how to separate $forall nin Bbb N$ and $b<frac1n$ is up to you, you can use a comma like I did, or a colon, or wrap $b<frac1n$ in parentheses).

share|cite|improve this answer

answered Mar 22 at 8:50

ArthurArthur

122k7122211

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  "..thus we have $f$ continuous for every $xin D$.." opposed to "..thus for every $xin D$ $f$ is continuous.." or do you regard worded and symbolised quantifications differently?
  $endgroup$
  – Alvin Lepik
  Mar 22 at 8:53
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @AlvinLepik I definitely regard worded and symbolic quantifiers differently. When using regular, human English, putting quantifiers after is natural, because that's how English semantics work.
  $endgroup$
  – Arthur
  Mar 22 at 8:54
  
  

  
  
  
  

add a comment | 

$begingroup$

Looks good to me.

Small notational nitpicks: It's Bbb N ("BlackBoard Bold") to make $Bbb N$.

And when using symbols to convey logic, like in your $b<frac1nforall nin Bbb N$, the quantifiers ($forall$ and $exists$) always come before whatever it is they modify.

So while one in English could say "$b$ is smaller than $frac1n$ for any natural number $n$", the symbolic statement must be $forall nin Bbb N, b<frac1n$ (exactly how to separate $forall nin Bbb N$ and $b<frac1n$ is up to you, you can use a comma like I did, or a colon, or wrap $b<frac1n$ in parentheses).

share|cite|improve this answer

answered Mar 22 at 8:50

ArthurArthur

122k7122211

$endgroup$

share|cite|improve this answer

answered Mar 22 at 8:50

ArthurArthur

122k7122211

answered Mar 22 at 8:50

ArthurArthur

122k7122211

answered Mar 22 at 8:50

ArthurArthur

122k7122211