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Pointwise Convergence

  1. Apr 9, 2013 #1
    Give an example of a sequence [itex]\{ f_n\}[/itex] of continuous functions defined on [0,1] such that [itex]\{ f_n\}[/itex] converges pointwise to the zero function on [0,1], but the sequence [itex]\{ \int^{1}_{0} f_n\}[/itex] is unbounded.

    I'm pretty lost on this one.
  2. jcsd
  3. Apr 9, 2013 #2


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    Hint: Try coming up with a sequence of functions such that ##f_n(x)## is only nonzero on the interval ##(0, 1/n)##. You will need to make them grow "taller" and "narrower" as ##n## increases.
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