i have a question about uniform convergence. definition of uniform convergence is given , there exists such that for all and for all , we have . i was wondering why the pointwise convergence does not imply uniform convergence. i know that for pointwise convergence, depending on , you have to find a different to have . but for each , will eventually be within the distance of for for some natural number so if you choose max of such and call it , then can we say for given , for all x and for all ?