some confusion about supremum and infimum












3












$begingroup$



Let $a_{mn}$ be a double array or real numbers. Define $$A=liminf_{nrightarrow infty} limsup_{mrightarrowinfty}a_{mn}\B=limsup_{nrightarrow infty} liminf_{mrightarrowinfty}a_{mn},$$



Then which one is true?




  1. $Ale B$

  2. $Age B $




My attempt :



I think option $1$ that $Ale B$ is true because the supremum is always greater than then infimum.



Is this correct ? Any hints/solution would be appreciated.










share|cite|improve this question











$endgroup$

















    3












    $begingroup$



    Let $a_{mn}$ be a double array or real numbers. Define $$A=liminf_{nrightarrow infty} limsup_{mrightarrowinfty}a_{mn}\B=limsup_{nrightarrow infty} liminf_{mrightarrowinfty}a_{mn},$$



    Then which one is true?




    1. $Ale B$

    2. $Age B $




    My attempt :



    I think option $1$ that $Ale B$ is true because the supremum is always greater than then infimum.



    Is this correct ? Any hints/solution would be appreciated.










    share|cite|improve this question











    $endgroup$















      3












      3








      3





      $begingroup$



      Let $a_{mn}$ be a double array or real numbers. Define $$A=liminf_{nrightarrow infty} limsup_{mrightarrowinfty}a_{mn}\B=limsup_{nrightarrow infty} liminf_{mrightarrowinfty}a_{mn},$$



      Then which one is true?




      1. $Ale B$

      2. $Age B $




      My attempt :



      I think option $1$ that $Ale B$ is true because the supremum is always greater than then infimum.



      Is this correct ? Any hints/solution would be appreciated.










      share|cite|improve this question











      $endgroup$





      Let $a_{mn}$ be a double array or real numbers. Define $$A=liminf_{nrightarrow infty} limsup_{mrightarrowinfty}a_{mn}\B=limsup_{nrightarrow infty} liminf_{mrightarrowinfty}a_{mn},$$



      Then which one is true?




      1. $Ale B$

      2. $Age B $




      My attempt :



      I think option $1$ that $Ale B$ is true because the supremum is always greater than then infimum.



      Is this correct ? Any hints/solution would be appreciated.







      real-analysis






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited Dec 15 '18 at 7:50









      Jonathan

      16312




      16312










      asked Dec 15 '18 at 6:02









      jasminejasmine

      1,781417




      1,781417






















          1 Answer
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          6












          $begingroup$

          Of course the supremum is always greater than or equal to the infimum. However, if one has mixtures of $limsup$ and $liminf$, it is indecisive which one is greater. For example, let $$
          a_{mn} = (-1)^n (1-frac{1}{2^m}).
          $$
          Then we have
          $$
          1=liminf_{mtoinfty}limsup_{ntoinfty}a_{mn} >limsup_{mtoinfty}liminf_{ntoinfty}a_{mn} =-1
          $$
          while
          $$
          -1=liminf_{ntoinfty}limsup_{mtoinfty}a_{mn}<limsup_{ntoinfty}liminf_{mtoinfty}a_{mn}=1.
          $$






          share|cite|improve this answer









          $endgroup$













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            $begingroup$

            Of course the supremum is always greater than or equal to the infimum. However, if one has mixtures of $limsup$ and $liminf$, it is indecisive which one is greater. For example, let $$
            a_{mn} = (-1)^n (1-frac{1}{2^m}).
            $$
            Then we have
            $$
            1=liminf_{mtoinfty}limsup_{ntoinfty}a_{mn} >limsup_{mtoinfty}liminf_{ntoinfty}a_{mn} =-1
            $$
            while
            $$
            -1=liminf_{ntoinfty}limsup_{mtoinfty}a_{mn}<limsup_{ntoinfty}liminf_{mtoinfty}a_{mn}=1.
            $$






            share|cite|improve this answer









            $endgroup$


















              6












              $begingroup$

              Of course the supremum is always greater than or equal to the infimum. However, if one has mixtures of $limsup$ and $liminf$, it is indecisive which one is greater. For example, let $$
              a_{mn} = (-1)^n (1-frac{1}{2^m}).
              $$
              Then we have
              $$
              1=liminf_{mtoinfty}limsup_{ntoinfty}a_{mn} >limsup_{mtoinfty}liminf_{ntoinfty}a_{mn} =-1
              $$
              while
              $$
              -1=liminf_{ntoinfty}limsup_{mtoinfty}a_{mn}<limsup_{ntoinfty}liminf_{mtoinfty}a_{mn}=1.
              $$






              share|cite|improve this answer









              $endgroup$
















                6












                6








                6





                $begingroup$

                Of course the supremum is always greater than or equal to the infimum. However, if one has mixtures of $limsup$ and $liminf$, it is indecisive which one is greater. For example, let $$
                a_{mn} = (-1)^n (1-frac{1}{2^m}).
                $$
                Then we have
                $$
                1=liminf_{mtoinfty}limsup_{ntoinfty}a_{mn} >limsup_{mtoinfty}liminf_{ntoinfty}a_{mn} =-1
                $$
                while
                $$
                -1=liminf_{ntoinfty}limsup_{mtoinfty}a_{mn}<limsup_{ntoinfty}liminf_{mtoinfty}a_{mn}=1.
                $$






                share|cite|improve this answer









                $endgroup$



                Of course the supremum is always greater than or equal to the infimum. However, if one has mixtures of $limsup$ and $liminf$, it is indecisive which one is greater. For example, let $$
                a_{mn} = (-1)^n (1-frac{1}{2^m}).
                $$
                Then we have
                $$
                1=liminf_{mtoinfty}limsup_{ntoinfty}a_{mn} >limsup_{mtoinfty}liminf_{ntoinfty}a_{mn} =-1
                $$
                while
                $$
                -1=liminf_{ntoinfty}limsup_{mtoinfty}a_{mn}<limsup_{ntoinfty}liminf_{mtoinfty}a_{mn}=1.
                $$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 15 '18 at 7:11









                SongSong

                14.1k1633




                14.1k1633






























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