Exercise 7.3

Question

Exercise 3: 
    Construct sequences $\{f_n\}$, $\{g_n\}$ which converge uniformly on some set $E$, but such that $\{f_ng_n\}$ does not converge uniformly on $E$ (of course, $\{f_ng_n\}$ must converge on $E$).

analambanomenos · Accepted Answer

Let $f_n(x)=x^{-1}+n^{-1}$ on $(0,1)$. Then $f_n$ converges uniformly to $x^{-1}$ on $(0,1)$, and $$f_n^2(x)=\frac{1}{x^2}+\frac{2}{nx}+\frac{1}{n^2}$$ converges to $x^{-2}$ on $(0,1)$, but
    not uniformly, since the difference $f_n^2(x)-x^{-2}=2/(nx)+1/n^2$ is arbitrarily large as $x\rightarrow 0$.

Exercise 7.3

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Exercise 7.3

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