I have the following proof in my notes: ( C [ 0 , 1 ] , ‖ ⋅ ‖ 1 ) is not completeConsider the sequence of functions f n ( t ) = { 1 t , 1 / n ≤ t ≤ 1 n , 0 ≤ t ≤ 1 n It can easily be shown that it is Cauchy in ‖ ⋅ ‖ 1 So we have that ∀ ε ∃ N ε such that ‖ f n − f ‖ ∞ ≤ ε ∀ m , n &gt; N ε Suppose f n converges, Then ∃ f ∈ C [ 0 , 1 ] such that ‖ f n − f ‖ 1 → 0.Then f n k → f a.e in [ 0 , 1 ] f ( t ) = lim k → ∞ f n k ( t ) = 1 t a.e in [ 0 , 1 ]. Absurd.I'm just failing to understand the last line. Can someone please explain1. what is the absurd?2. And how does pointwise convergence follows from a.e convergence (last line). Am I supposed to understand this f n k → f a.e in [ 0 , 1 ] as convergence in the real numbers, with the absolute value for every fixed t ( | f n k ( t ) − f ( t ) | a.e in [ 0 , 1 ]) ?

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I have the following proof in my notes:  (  C  [  0  ,  1  ]  ,  ‖  ⋅      ‖    1    ) is not completeConsider the sequence of functions      f    n    (  t  )  =      {                                        1                          t                                                ,                          1                      /                    n          ≤          t          ≤          1                                                  n                                    ,                          0          ≤          t          ≤                      1            n                                  It can easily be shown that it is Cauchy in   ‖  ⋅      ‖    1   So we have that   ∀  ε  ∃      N    ε   such that   ‖      f    n    −  f      ‖    ∞    ≤  ε  ∀  m  ,  n  &amp;gt;      N    ε  Suppose       f    n   converges, Then   ∃  f  ∈  C  [  0  ,  1  ] such that   ‖      f    n    −  f      ‖    1    →  0.Then       f                  n        k              →  f   a.e in  [  0  ,  1  ]  f  (  t  )  =      lim          k      →      ∞            f                  n        k              (  t  )  =      1          t         a.e in  [  0  ,  1  ]. Absurd.I&#039;m just failing to understand the last line. Can someone please explain1. what is the absurd?2. And how does pointwise convergence follows from a.e convergence (last line). Am I supposed to understand this       f                  n        k              →  f   a.e in  [  0  ,  1  ] as convergence in the real numbers, with the absolute value for every fixed t (      |        f                  n        k              (  t  )  −  f  (  t  )      |     a.e in  [  0  ,  1  ]) ?

sleuteleni7 · Accepted Answer

1. The limit function   f does not belong to       (    C    (    [    0    ,    1    ]    )    ,    ‖    ⋅          ‖                        ℓ          1                      )    , creating the contradiction since   f was supposed to belong to this space. More precisely,   f is equal to   1      /        t   a.e, but by continuity,  f  (  0  )  =      lim          t      →      0        f  (  t  )  =  +  ∞  ,and therefore   f may not be continuous at 0, since continuous functions on [0,1] are in particular finite at 0, a contradiction.

Makayla Boyd · Accepted Answer

2. a.e convergence means precisely that the pointwise convergence is valid a.e.

I have the following proof in my notes: ( C [ 0 , 1 ] , <mo fence="fals

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