Consider the sequence of functions on $(0,\infty)$ given by


Does the sequence converge pointwise? Does it converge uniformly?

My attempt: I can see that as $n\rightarrow\infty$, $f_n(x)\rightarrow 0$ for each $x$, so it must converge pointwise. For uniform convergence, we need that

$\sup_x\left|\frac{nx}{1+n^2x^2}-0\right|\rightarrow 0$ as $n\rightarrow\infty$. This appears to be the case, but am I wrong?

Any help appreciated!


Note that, for all $n$, $$f_n(x)=\frac{nx}{1+n^2x^2}\implies f_n(1/n)=\frac{1}{2}$$ As a result, $f_n$ cannot tend uniformly to $0$.


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