Suppose that $f_n,f:X\to Y$, that $(f_n)\to f$ point-wise, and $f_n$ and $f$ are bounded for all $n\in\Bbb N$. Then we have that
$$\lim_{n\to\infty}(f(x)-f_n(x))=0,\quad\forall x\in X$$
Because $f_n$ and $f$ are bounded then $\|f_n\|_\infty$ and $\|f\|_\infty$ are finite, thus
$$\lim_{n\to\infty}\|f-f_n\|_\infty=\|f-\lim_{n\to\infty}f_n\|_\infty=\|0\|_\infty=0$$
because $\|{\cdot}\|_\infty$ is continuous. But the above "proof" is false because if would imply that point-wise convergence of bounded functions imply uniform convergence, what doesnt seems true regarding Dini's theorem.
But I cant see the mistake, can someone enlighten me please? Thank you.
The mistake is $\lim_{n \to \infty} ||f-f_n||_\infty = ||f-\lim_n f_n||_\infty$. You have to be careful when you say things like " $||\cdot||_\infty$ is continuous." That is true, but that doesn't apply here since the limit we are taking is a pointwise limit. $||\cdot||_\infty$ is continuous means that if $\lim_n f_n = f$ in the uniform metric (i.e. $f_n \to f$ uniformly), then $\lim_n ||f_n||_\infty = ||f||_\infty$.