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$\mathbb{R}$ is the domain. Let $$f_n(x) = \frac{4n}{n+x^2}$$

As $n$ becomes large the $x^2$ term becomes insignificant and the function converges to $4$ pointwise.

Now it seems to me that no matter what $N$ I choose, $$|f_n(x) - f(x)| < \epsilon$$

doesn't hold when $n \ge N$ for every $x$ as I can choose 'a very large $x$' to such that the $n$ values become insignificant and $f_n(x)$ approaches $0$. Is that correct? And how would I put this formally?

csss
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1 Answers1

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$$|f_n(x)-f(x)|=\left|\frac{4n}{n+x^2}-4\right|=\left|\frac{-x^2}{n+x^2}\right|<\epsilon\;\;\;\underline{\color{red}{and\;\forall\,x\in\Bbb R}}\iff$$

$$x^2<\epsilon n+\epsilon x^2\iff n>\frac{x^2(1-\epsilon)}{\epsilon}$$

and the above confirms your thought that choosing $\;x\;$ pretty large and no matter what $\;\epsilon >0\;$ is, we won't get the above inequality

DonAntonio
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