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I got $$ f_n(x) = \frac{3x^2 (\frac 1 {n^2} + x^2) - 2x^4} { (\frac 1 {n^2} +x^2) ^2} $$ on $[-1,1]$.

I have already proven that it converges pointwise to $$ f(x) := \begin{cases} 1 & x \neq 0 \\ 0 & x = 0 \end{cases} $$

Can someone tell me if this sequence converges uniform to $f$, too ?

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Is the pointwise limit continuous? –  David Mitra Jan 20 '13 at 17:41
    
Alternatively, what is $f_n(1/n)$? –  David Mitra Jan 20 '13 at 17:47

1 Answer 1

up vote 2 down vote accepted

No, if you calculated correctly the limit function, then $f_n$ is not uniformly convergent. as your $f$ is not continous.

Here is a result you may know: if $f_n$ are all continous on a domain $D$ and if they converge uniformly on $D$, the limit function must be continuos.

your $f_n$ are all continous but limit function is not.

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Yes, sure :) Thanks. –  André Jan 20 '13 at 17:46
    
@André you are welcome! –  Une Femme Douce Jan 20 '13 at 17:47

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