Rudin 2.10 (b) Example












4












$begingroup$


Let $A$ be the set of real numbers such that $0 < x leq 1$ . For every $x in A$, let $E_{x}$ be the set of real numbers $y$ such that $0< y< x$. Then





  1. $bigcap_{x in A} E_{x}$ is empty.


The Proof provided in textbook:
We note that for every $y > 0$, $y notin E_{x}$ if $x < y$. Hence $y notin bigcap_{x epsilon A} E_{x}$



I didn't understand the proof & also here is my understanding.



Counter Argument-1: We can always find a real number between every $(0,x)$ where x is $x > 0$. So, It can not be empty.



Counter Argument-2: This is somehow similar to Nested interval property, So, it can not be empty.



Please explain how rudin got this result ?






real-analysis set-theory







share|cite|improve this question









New contributor




Mahendra Reddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    He is saying that for any $y$, you can find an $x$ in the interval such that $xleq y$ and so $y$ is not in $E_x$ for that $x$ and so it cannot be in the intersection of all the $E_x$. Since $y$ was arbitrary, the intersection must be empty.
    $endgroup$
    – zbrads2
    4 hours ago










  • $begingroup$
    In addition to Martin's answer, the Nested Interval Property requires closed intervals.
    $endgroup$
    – Lucas Corrêa
    4 hours ago
















4












$begingroup$


Let $A$ be the set of real numbers such that $0 < x leq 1$ . For every $x in A$, let $E_{x}$ be the set of real numbers $y$ such that $0< y< x$. Then





  1. $bigcap_{x in A} E_{x}$ is empty.


The Proof provided in textbook:
We note that for every $y > 0$, $y notin E_{x}$ if $x < y$. Hence $y notin bigcap_{x epsilon A} E_{x}$



I didn't understand the proof & also here is my understanding.



Counter Argument-1: We can always find a real number between every $(0,x)$ where x is $x > 0$. So, It can not be empty.



Counter Argument-2: This is somehow similar to Nested interval property, So, it can not be empty.



Please explain how rudin got this result ?






real-analysis set-theory







share|cite|improve this question









New contributor




Mahendra Reddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    He is saying that for any $y$, you can find an $x$ in the interval such that $xleq y$ and so $y$ is not in $E_x$ for that $x$ and so it cannot be in the intersection of all the $E_x$. Since $y$ was arbitrary, the intersection must be empty.
    $endgroup$
    – zbrads2
    4 hours ago










  • $begingroup$
    In addition to Martin's answer, the Nested Interval Property requires closed intervals.
    $endgroup$
    – Lucas Corrêa
    4 hours ago














4












4








4





$begingroup$


Let $A$ be the set of real numbers such that $0 < x leq 1$ . For every $x in A$, let $E_{x}$ be the set of real numbers $y$ such that $0< y< x$. Then





  1. $bigcap_{x in A} E_{x}$ is empty.


The Proof provided in textbook:
We note that for every $y > 0$, $y notin E_{x}$ if $x < y$. Hence $y notin bigcap_{x epsilon A} E_{x}$



I didn't understand the proof & also here is my understanding.



Counter Argument-1: We can always find a real number between every $(0,x)$ where x is $x > 0$. So, It can not be empty.



Counter Argument-2: This is somehow similar to Nested interval property, So, it can not be empty.



Please explain how rudin got this result ?






real-analysis set-theory







share|cite|improve this question









New contributor




Mahendra Reddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




Let $A$ be the set of real numbers such that $0 < x leq 1$ . For every $x in A$, let $E_{x}$ be the set of real numbers $y$ such that $0< y< x$. Then





  1. $bigcap_{x in A} E_{x}$ is empty.


The Proof provided in textbook:
We note that for every $y > 0$, $y notin E_{x}$ if $x < y$. Hence $y notin bigcap_{x epsilon A} E_{x}$



I didn't understand the proof & also here is my understanding.



Counter Argument-1: We can always find a real number between every $(0,x)$ where x is $x > 0$. So, It can not be empty.



Counter Argument-2: This is somehow similar to Nested interval property, So, it can not be empty.



Please explain how rudin got this result ?






real-analysis set-theory




real-analysis set-theory






share|cite|improve this question









New contributor




Mahendra Reddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Mahendra Reddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 4 hours ago









Lucas Corrêa

1,6151421




1,6151421






New contributor




Mahendra Reddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 5 hours ago









Mahendra ReddyMahendra Reddy

212




212




New contributor




Mahendra Reddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Mahendra Reddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Mahendra Reddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • $begingroup$
    He is saying that for any $y$, you can find an $x$ in the interval such that $xleq y$ and so $y$ is not in $E_x$ for that $x$ and so it cannot be in the intersection of all the $E_x$. Since $y$ was arbitrary, the intersection must be empty.
    $endgroup$
    – zbrads2
    4 hours ago










  • $begingroup$
    In addition to Martin's answer, the Nested Interval Property requires closed intervals.
    $endgroup$
    – Lucas Corrêa
    4 hours ago


















  • $begingroup$
    He is saying that for any $y$, you can find an $x$ in the interval such that $xleq y$ and so $y$ is not in $E_x$ for that $x$ and so it cannot be in the intersection of all the $E_x$. Since $y$ was arbitrary, the intersection must be empty.
    $endgroup$
    – zbrads2
    4 hours ago










  • $begingroup$
    In addition to Martin's answer, the Nested Interval Property requires closed intervals.
    $endgroup$
    – Lucas Corrêa
    4 hours ago
















$begingroup$
He is saying that for any $y$, you can find an $x$ in the interval such that $xleq y$ and so $y$ is not in $E_x$ for that $x$ and so it cannot be in the intersection of all the $E_x$. Since $y$ was arbitrary, the intersection must be empty.
$endgroup$
– zbrads2
4 hours ago




$begingroup$
He is saying that for any $y$, you can find an $x$ in the interval such that $xleq y$ and so $y$ is not in $E_x$ for that $x$ and so it cannot be in the intersection of all the $E_x$. Since $y$ was arbitrary, the intersection must be empty.
$endgroup$
– zbrads2
4 hours ago












$begingroup$
In addition to Martin's answer, the Nested Interval Property requires closed intervals.
$endgroup$
– Lucas Corrêa
4 hours ago




$begingroup$
In addition to Martin's answer, the Nested Interval Property requires closed intervals.
$endgroup$
– Lucas Corrêa
4 hours ago










2 Answers
2






active

oldest

votes


















3












$begingroup$

Note that $bigcap_{xin A}E_xsubset(0,infty)$. If $yin bigcap_{xin A}E_xsubset(0,infty)$, this means that $y<x$ for all $x>0$. As Rudin says, this is a contradiction because given $y>0$, you can always find $x$ with $0<x<y$ (for instance, $x=y/2$).



Regarding your comment, the nested interval property is about compact sets. These are open and not closed.






share|cite|improve this answer











$endgroup$





















    0












    $begingroup$

    Note that $xnotin E_x$ for every $x$






    share|cite|improve this answer








    New contributor




    Andreé Ríos is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






    $endgroup$













    • $begingroup$
      Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
      $endgroup$
      – dantopa
      3 hours ago












    Your Answer








    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
    });


    }
    });






    Mahendra Reddy is a new contributor. Be nice, and check out our Code of Conduct.










    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3202687%2frudin-2-10-b-example%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    Note that $bigcap_{xin A}E_xsubset(0,infty)$. If $yin bigcap_{xin A}E_xsubset(0,infty)$, this means that $y<x$ for all $x>0$. As Rudin says, this is a contradiction because given $y>0$, you can always find $x$ with $0<x<y$ (for instance, $x=y/2$).



    Regarding your comment, the nested interval property is about compact sets. These are open and not closed.






    share|cite|improve this answer











    $endgroup$


















      3












      $begingroup$

      Note that $bigcap_{xin A}E_xsubset(0,infty)$. If $yin bigcap_{xin A}E_xsubset(0,infty)$, this means that $y<x$ for all $x>0$. As Rudin says, this is a contradiction because given $y>0$, you can always find $x$ with $0<x<y$ (for instance, $x=y/2$).



      Regarding your comment, the nested interval property is about compact sets. These are open and not closed.






      share|cite|improve this answer











      $endgroup$
















        3












        3








        3





        $begingroup$

        Note that $bigcap_{xin A}E_xsubset(0,infty)$. If $yin bigcap_{xin A}E_xsubset(0,infty)$, this means that $y<x$ for all $x>0$. As Rudin says, this is a contradiction because given $y>0$, you can always find $x$ with $0<x<y$ (for instance, $x=y/2$).



        Regarding your comment, the nested interval property is about compact sets. These are open and not closed.






        share|cite|improve this answer











        $endgroup$



        Note that $bigcap_{xin A}E_xsubset(0,infty)$. If $yin bigcap_{xin A}E_xsubset(0,infty)$, this means that $y<x$ for all $x>0$. As Rudin says, this is a contradiction because given $y>0$, you can always find $x$ with $0<x<y$ (for instance, $x=y/2$).



        Regarding your comment, the nested interval property is about compact sets. These are open and not closed.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 4 hours ago

























        answered 4 hours ago









        Martin ArgeramiMartin Argerami

        130k1184185




        130k1184185























            0












            $begingroup$

            Note that $xnotin E_x$ for every $x$






            share|cite|improve this answer








            New contributor




            Andreé Ríos is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.






            $endgroup$













            • $begingroup$
              Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
              $endgroup$
              – dantopa
              3 hours ago
















            0












            $begingroup$

            Note that $xnotin E_x$ for every $x$






            share|cite|improve this answer








            New contributor




            Andreé Ríos is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.






            $endgroup$













            • $begingroup$
              Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
              $endgroup$
              – dantopa
              3 hours ago














            0












            0








            0





            $begingroup$

            Note that $xnotin E_x$ for every $x$






            share|cite|improve this answer








            New contributor




            Andreé Ríos is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.






            $endgroup$



            Note that $xnotin E_x$ for every $x$







            share|cite|improve this answer








            New contributor




            Andreé Ríos is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.









            share|cite|improve this answer



            share|cite|improve this answer






            New contributor




            Andreé Ríos is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.









            answered 4 hours ago









            Andreé RíosAndreé Ríos

            1




            1




            New contributor




            Andreé Ríos is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.





            New contributor





            Andreé Ríos is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.






            Andreé Ríos is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.












            • $begingroup$
              Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
              $endgroup$
              – dantopa
              3 hours ago


















            • $begingroup$
              Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
              $endgroup$
              – dantopa
              3 hours ago
















            $begingroup$
            Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
            $endgroup$
            – dantopa
            3 hours ago




            $begingroup$
            Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
            $endgroup$
            – dantopa
            3 hours ago










            Mahendra Reddy is a new contributor. Be nice, and check out our Code of Conduct.










            draft saved

            draft discarded


















            Mahendra Reddy is a new contributor. Be nice, and check out our Code of Conduct.













            Mahendra Reddy is a new contributor. Be nice, and check out our Code of Conduct.












            Mahendra Reddy is a new contributor. Be nice, and check out our Code of Conduct.
















            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3202687%2frudin-2-10-b-example%23new-answer', 'question_page');
            }
            );

            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







            -

            Popular posts from this blog

            數位音樂下載

            Image
            數位音樂下載 為非以「實體」方式來販賣音樂的相關產品，而使用數位格式如mp3、AAC等文件格式來進行銷售販賣的一种音乐产业形式。著名的數位音樂下載服務商有iTunes Store、RecoChoku、Google Play音乐和微软Zune等。数字音乐下载在二十一世纪初一度风靡全球， [1] [2] 使得传统唱片产业走向衰落，但其自身也在2010年代流媒体音乐服务如Apple Music、Spotify等的挤压下迅速失去市场份额。 [3] [4] 数字音乐下载有便于携带、传播迅速、制作简单成本低等优点，与流媒体音乐相比也能更好保障音乐人权益； [5] 但它同时很可能带来侵犯版权和滥用的情形， [6] 也给音乐产业带来了诸多不利影响。数字音乐下载为音乐人和乐迷提供了更加便捷直接的交流渠道，也一定程度上推动了网络音乐人的出现和独立音乐的迅速发展。 [7] 目录 1 历史 1.1 技术积淀（1987-2000） 1.2 出现与兴起（2000-2004） 1.3 唱片市场的颠覆者（2004-2007） 1.4 巅峰和衰落（2005-2013） 1.5 摇摇欲坠的现状与不可预知的明天（2014至今） 2 主要文件格式 2.1 PCM编码格式 2.1.1 WAV 2.1.2 APE 2.1.3 FLAC 2.2 苹果推出的无损文件格式 2.2.1 AIFF 2.2.2 ALAC 2.3 有损压缩格式 2.3.1 MP3 2.3.2 AAC 3 主要市场 3.1 iTunes Store 3.2 Google Play音乐商店 3.3 亚马逊音乐商店 3.4 部分其他音乐商店 4 影响与评价 4.1 对音乐产业的影响 4.2 对科技行业的影响 4.3 正面评价 4.4 负面评价 4.4.1 侵权问题 4.4.2 经营模式不明晰 4.4.3 不适应移动时代 4.5 文化现象 5 延伸阅读 6 参考资料 历史 苹果iTunes+iPod的软硬件搭配让数字音...
            Read more

            When can things happen in Etherscan, such as the picture below?

            Image
            2 I don't know why balance of the address is 0. when can this situation? etherscan balances share | improve this question asked yesterday Ru Hi Ru Hi 11 1 New contributor Ru Hi is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct. add a comment  |  ...
            Read more

            格利澤436b

            Image
            格利泽436b 太陽系外行星 太陽系外行星列表 艺术家笔下的格利泽436b 母恆星 母恆星 Gliese 436 星座 獅子座 赤經 ( α ) 11 h  42 m  11.0941 s [1] 赤緯 ( δ ) +26° 42′ 23.652″ [1] 距離 33.4 ly (10.2 pc) 光譜類型 M2.5 V [1] 軌道參數 半长轴 ( a ) 0.0291±0.0004 [2] AU 軌道離心率 ( e ) 0.150±0.012 [2] 公轉週期 ( P ) 2.643904±0.000005 [3] d (0.00723849 y) 軌道傾角 ( i ) 85.8 +0.21 −0.25 [3] ° 角距 ( θ ) 2.794 mas 近星點時間 ( T 0 ) 2,451,551.716 ±0.01 JD 半振幅 ( K ) 18.68±0.8 m/s 物理性质 质量 ( m ) 22.2±1.0 [2] M ⊕ 半徑 ( r ) 4.327±0.183 [2] [4] R ⊕ 密度 ( ρ ) 1.51 g cm -3 表面重力 ( g ) 1.18 g 溫度 ( T ) 712±36 [2] K 發現 發現時間 2004 發現者 巴特勒、沃格特、 馬西 et al. 發現方法 徑向速度、凌日法 發現地點 美國加利福尼亞州 發表論文 出版 其他名稱 Ross 905 b, GJ 436 b, [5] LTT 13213 b, GCTP 2704.10 b, LHS 310, AC+27:28217 b, Vyssotsky 616 b, HIP 57087 b, GEN# +9.80120068 b, LP 319-75 b, G 121-7 b, LSPM J1142+2642 b, 1RXS J114211.9+264328 b, ASCC 683818 b, G 147-68 b, UCAC2 41198281 b, BP...
            Read more