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If we consider the space $L^\infty ([0,1],\,dx)$, why is it that both monotone and dominated convergence fail?

My first take on the problem was to consider the characteristic function $f_n$ of $[0,1-\frac1n]$, but I wasn't able to proceed from there.

Thank you.

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    $f_n$ is increasing to the constant $1$ (hence dominated by it), and converges almost everywhere to this function. Since $|1-f_n|=\mathbf 1_{\left[-\frac 1n,1\right]}$, we have $\lVert 1-f_n\rVert_{\infty}=1$ and we can't have the convergence on $L^{\infty}$-norm.2011-11-21
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    Is it homework? Did you try what happens to $f_n(x)=x^n$?2011-11-21

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