i have a question about uniform convergence. definition of uniform convergence is given $\epsilon >0$, there exists $N$ such that for all $x$ and for all $n \geq N$, we have $|f_n (x) -f(x)|<\epsilon$. i was wondering why the pointwise convergence does not imply uniform convergence. i know that for pointwise convergence, depending on $x$, you have to find a different $N$ to have $|f(x)-f_n(x)| < \epsilon$. but for each $x$, $f_n$ will eventually be within the distance of $\epsilon$ for $n \geq N_x$ for some natural number $N_x$ so if you choose max of such $N_x$ and call it $M$, then can we say for given $\epsilon >0$, for all x and for all $n \geq M$ $|f_n(x)-f(x)|< \epsilon$?
2. It might be that $\sup N_x=\infty$. Just consider $f_n(x)=1$ if $x\in [n,n+1]$ and 0 otherwise. Then $f_n\to 0$, but it isn't uniform.
3. max of $N_x$?
it is possible that $N_x$ is not bounded over all x.