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I came up with the counterexample $f_n = 1/n$ such that it converges uniformly to $f=0$ but I am not sure that its a good one because $1/f$ is not defined. Do you think its a good counterexample? Can anyone come up with another one?

Thanks!

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    Better pack all [your exercise problems for the same topic](http://math.stackexchange.com/questions/253410/if-f-n-rightarrow-f-uniformly-andf-is-bounded-then-prove-that-f-n2-r) in one question.2012-12-07

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Since the limit of $\frac1{f_n}$ does not even exist, I would consider this a fine counterexmple. On the other hand, what about $f_n\colon (0,1)\to\mathbb R$, $x\mapsto x+\frac1n$? Then the reciprocal of the limit does exist, but the convergence is not uniform.

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    @UH1: It also isn't hard to show directly that for each $n$ there exists $x$ such that \left|\dfrac{1}{f(x)}−\dfrac{1}{f_n(x)}\right|>1.2012-12-08