$$f_n(x)=\frac1{1+nx^2}$$
a) Find the pointwise limit.
b) Prove $f_n(x)$ is not uniformly convergent.
Sequence of Functions $\frac{1}{1+nx^2}$
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sequences-and-series
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0On what interval are you trying to show that it fails to be UC? It is UC on any closed interval that excludes $x=0$. – 2017-02-28
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0Consider sharing what all you have tried. MSE will be able to help you better then. – 2017-02-28
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2Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – 2017-02-28
2 Answers
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Hint
a) Take the limit for $n \to \infty$ and consider the cases $x = 0$ and $x \ne 0$ separately.
b) Either go back to the definition or use the result from part a), where you should find that the limit function isn't continuous.
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Hint: a) pointwise limit is a discontinuous, b) if the sequence of continuous functions converges uniformly, limiting function is continuous