My instructor briefly discussed a result in lecture that I need help with. Here was the set up.We assumed that { f n } is an orthonormal system in L 2 ( 0 , 1 ) and we supposed that there exists an M &gt; 0 such that | f n ( x ) | ≤ M a.e for all n ∈ N . Furthermore, we let { c n } be a sequence of real numbers such that ∑ n = 1 ∞ c n f n cnfn converges a.e.They said that it was "clear" that lim n → ∞ c n = 0.This is not clear or intuitive for me.After reading through the page, I am still a bit confused. Can someone put together a working proof to clear things up?

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My instructor briefly discussed a result in lecture that I need help with. Here was the set up.We assumed that   {      f    n    } is an orthonormal system in       L    2    (  0  ,  1  ) and we supposed that there exists an   M  &amp;gt;  0 such that       |        f    n    (  x  )      |    ≤  M a.e for all   n  ∈      N  . Furthermore, we let   {      c    n    } be a sequence of real numbers such that       ∑          n      =      1              ∞            c    n        f    n  cnfn converges a.e.They said that it was &quot;clear&quot; that      lim          n      →      ∞            c    n    =  0.This is not clear or intuitive for me.After reading through the page, I am still a bit confused. Can someone put together a working proof to clear things up?

Shawn Castaneda · Accepted Answer

Suppose the sum of an infinite series is     S   ,and we denote by         S    n     the     n    −−th element of the sequence of partial sums of the series. By definition, the series converges iff the limit of         S    n     exists finitely ( and in this case         S    n    →  S  ) , so:  S  =      ∑          n      =      1        ∞        a    n      ,          S    n    :=      ∑          k      =      1        n        a    k      ⟹        a    n    =      S    n    −      S          n      −      1            →          n      →      ∞                              S                    (arith. of limits)              −  S  =  0and we have the basic necessary condition for convergence of infinite series.In your case you have         |        c    n        f    n        |        →          n      →      ∞            0   (no matter whether real of complex numbers), but     {      f    n    }   is a bounded sequence...! Finish the argument.

My instructor briefly discussed a result in lecture that I need help with. Here was the set up. We

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