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From the joint image, how does $f_n \rightarrow 0$ pointwise on $\mathbb{R}$?

For instance, I can't find an integer $A$ such that $n>A \Longrightarrow |f_n(x)|\le 1/2, \forall x\in \mathbb{R}$ ?

Smilia
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  • Your $A$ can not be found (for ech $n$ there will always be an $x$ such that $f_n(x)=1$). And that's ok because $f_n$ converges only pointwise to zero but not uniformly. – Stefano Mar 16 '17 at 11:40
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    In pointwise convergence, you search an $A$ for each $x$ (they can be different!), not an $A$ for all $x$ – Tryss Mar 16 '17 at 12:22

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For $n>x$, we have $f_n(x)=1_{[n,n+1]}(x)=0$. So for any $x$, as $n \to \infty$, $f_n(x) \to 0$.

Sanderr
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