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