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Can you give an example of a sequence of continuous functions $f_n:[0,1]\to [0,1]$, such that $f_n\to 0$ pointwise and there is no subsequence $(f_{n_k})$ for which $\frac 1 m\sum_{k=1}^{m}f_{n_k}$ tends to zero uniformly?

I think it's the same as asking whether the Banach space of continuous real valued functions has the "weak Banach-Saks property", but I was unable to find out the answer.

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    @JesseMadnick: not much. I was hoping someone would just know the answer...2012-02-09
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    Did you try some triangular wave which increases frequency but such that $f_{n + 1}$ has the same peaks as $f_n$ but more of them?2012-02-09
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    [What have you tried so far?] *not much. I was hoping someone would just know the answer...*2012-02-09
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    It was somewhat rhetoric, @Didier. (if you are replying on my comment)2012-02-09
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    @Jonas: Your comment is standard.2012-02-09

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