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Let $(f_n)_{n\in\mathbb N}$ be a sequence of measurable functions on $[0,1]$ with $\forall n\in\mathbb N, \lvert f_n(x) \rvert < +\infty$ a.e. Show there exists a sequence $(c_n)_{n\in\mathbb N}$ of positive real numbers s.t. $f_n(x)/c_n \rightarrow 0$ for almost every $x$ in $[0,1]$. Hint: use the borel-cantelli lemma- pick the sequence $c_n$ so that $$m[x \in [0,1]:\frac{|f_n(x)|}{c_n}>1/n]<1/2^n$$ where m is the measure.


marked as duplicate by The Chaz 2.0, Amzoti, Henry T. Horton, Jim, Micah May 7 '13 at 3:02

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    $\begingroup$ Can you show us ideas you have had? $\endgroup$ – Kris Williams May 6 '13 at 17:06
  • $\begingroup$ I think because the borel cantelli lemma says that the set of points that belong to infinitely many measurable subsets of [0,1], is measurable and has measure zero, then the measure of the above set is 0 and thus c sub n must be a sequence of numbers that are each greater than 2^-n. $\endgroup$ – Real Anal May 6 '13 at 17:17
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    $\begingroup$ Dude seriously? You asked this question yesterday:math.stackexchange.com/questions/381780/…. $\endgroup$ – Quinn Culver May 7 '13 at 1:42
  • $\begingroup$ Voted to close as duplicate $\endgroup$ – The Chaz 2.0 May 7 '13 at 1:43