uniform convergence

**PRLM** · January 9th 2010, 09:27 PM

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 $n \geq M$ $|f_n(x)-f(x)|< \epsilon$ ?