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I am currently struggling with a very simple exercise, I guess.

I don't see why $$\lim_{n \to \infty} n\mathbb{1}_{(0,\frac{1}{n})}=0, $$ where $\mathbb{1}_{(0,\frac{1}{n})}$ is the indicator function.

Thanks

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  • $\begingroup$ On what variable does that indicator function is being applied? Perhaps I miss something trivial here, but the expression $\;\lim\limits_{n\to\infty} n\cdot f\;$ , with $\;f\;$ some function, doesn't make much sense unless we specify some variable for $\;f\;$ ... $\endgroup$ – DonAntonio May 12 '18 at 9:09
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Assuming you meant $\;n\cdot 1_{\left(0,\frac1n\right)}(x)\;$ , for any (some) fixed $\;x\in\Bbb R\;$, we get that there exists some $\;N\in\Bbb N\;$ s.t. for $\;n>N\implies 1_{\left(1,\frac1n\right)}(x)=0\;$ , and thus $\;n\cdot 1_{\left(1,\frac1n\right)}(x)=0\;$ for all $\;n>N\;$ , which of course means the limit is zero...

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