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!
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!
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.