Note that for x≥1, f_n(x) = nx/(1+nx^2) < nx/(nx^2) = 1/x ≤ 1 so each of the is bounded above by on so since is continuous on and is compact we have by the extreme value theorem that is bounded above on taking the greater of and the upper bound for on , upper bound for on . Therefore each one of the is bounded above on .

to find the pointwise limit of the , we see that for

this approaches 1/x as n\forwardarrow\infinity now for every n, , we have

n---->infinity lim f_n(x) = {0 if x=0, 1/x if x≠0}

this shows it is not bounded above on . It follows that the cannot converge uniformly