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Given a sequence of functions $f_n$ which is known to be pointwise convergent how would you go about showing that is not uniform convergent?

The particular example I'm working with is $f_n (x)=\frac {nx} {x^2+n^2}$ and I've tried using Theorem 7.9 from Rudin (uniform convergence $\iff M_n=(\sup |f_n(x)-f(x)|) \rightarrow 0$ as $n\rightarrow \infty$) but using that I found $M_n$ to be $\frac {\sqrt n} {1+n}$ by maximising $f_n$ but this gives the wrong result.

edit: $f:\mathbb R \rightarrow \mathbb R $, sorry about that

  • 0
    So, what makes you think it is not uniform convergent?2011-05-21
  • 1
    Are your examples defined on a real line, or at least an unbounded domain? Just notice that $f_n (x) = f_1 (x/n)$. So incresing $n$ has an effect of shifting the extreme point from 1 to $n$, which makes the convergence fail to be uniform. (But it enjoys a weaker but still sufficiently nice property for most circumstances that it converges uniformly on every compact subset of the domain.)2011-05-21
  • 0
    Your $M_n$ is wrong, try $f_n(n)$.2013-12-09

3 Answers 3