Define A n = { ( 1 2 − 1 2 n , 1 + 1 2 n ) if n is odd ( 1 2 n , 3 4 − 1 2 n ) if n is even Find lim inf n → ∞ A n and lim sup n → ∞ A n lim sup n → ∞ A n = ∩ n = 1 ∞ ( ∪ n ≤ N ) lim inf n → ∞ A n = ∪ n = 1 ∞ ( ∩ n ≤ N ) A 1 = ( 0 , 1 + 1 2 ) A 2 = ( 1 4 , 1 2 ) A 3 = ( 1 3 , 1 + 1 6 ) Can you help someone here i'm confusing

Question

Define      A    n    =      {                                        (                          1              2                        −                          1                              2                n                                      ,            1            +                          1                              2                n                                      )                                     if n is odd                                                   (                          1                              2                n                                      ,                          3              4                        −                          1                              2                n                                      )                                     if n is even                         Find       lim inf          n      →      ∞            A    n   and       lim sup          n      →      ∞            A    n        lim sup          n      →      ∞            A    n    =      ∩          n      =      1              ∞        (      ∪          n      ≤      N        )      lim inf          n      →      ∞            A    n    =      ∪          n      =      1              ∞        (      ∩          n      ≤      N        )      A    1    =      (    0    ,    1    +          1      2        )          A    2    =      (          1      4        ,          1      2        )          A    3    =      (          1      3        ,    1    +          1      6        )  Can you help someone here i&#039;m confusing

Alessandro Schmidt · Accepted Answer

A good way to think is like this:-1.Lim sup of a sequence of sets       A          n       is the set of all   x such that   x  ∈      A          n       for infinitely many   n.2. Lim inf of       A          n       is the set of all   x such that   x  ∈      A          n       for all but finitely many   n.you see that all   x  ∈  [      1    2    ,  1  ] . It lies in       A          n       for all odd   n. and all   x  ∈  (  0  ,      3    4    ) lies in       A          n       for infinitely many even integers   n.So   lim  sup      A          n        =  [      1    2    ,  1  ]  ∪  (  0  ,      3    4    )  =  (  0  ,  1  ]Now only   x such that   x  ∈  [      1    2    ,      3    4    ) lies in       A          n       for all but finitely many   n. So   lim  inf      A          n        =  [      1    2    ,      3    4    )

Define A n </msub> = { <mtable columnalign="left left" rowspaci

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