Question

THEOREM 205. Define the functions fr : [0, 1] + R by Sn(:1) = x" /n for n E N. The sequence n H Sn converges uniformly to the function f = 0, but the sequence n o fh does not converge to f' = 0. Note that the operations of taking a limit and taking a derivative do not necessarily commute.

Answer 1

Theorem? (205) - fritat ini [0, 1] - IR 1 fn (1) = 2 for nein. het Uniform convergence het SCR and for each 1 NEN; let fnis IR be a function. The sequence afa} is said to be uniformly convergent on s to a function f if for and t o there exusts a natural number n (depending on € but not on aes) for alt ny such that lin(a)-foolk f. for all nom. Proof is fon (1) = 2 lim forse) = lessons in Now , for all & E [0, 1] Ha (M) – f(x) = -0 Ź .. het Eyo. LLE - no Thus Ifa () - fla) <t for all na het m= {f}+1 Here m is a natural number that Only on E but not on a... depends

tot This proves that in converges uniformly to the function fro.. Boved I method: Mn-test: s pointurise het Lefn} converges pointuels Define Muc sub b-farm) -' f(a) then fri} converges uniformly to f if and only if lim Mu = 0 Here faran se LES i Ha cold a fl-1) = 211 Mua rup 1 = lim Mua lim to Thus {fr} converges uniformely foln) n-soo hoooo to fao.. farlo) 2:34 not lim 21 so txt[0,1) 1 1 :x=1 limfn' (d) = Take fron tfal (01). — .f'60) = 12-01 = 1024.? Since azo)

din) = xh put drain) LE 7 xhl LE log on-l <loge = (h-1) loga < loge >> - (n-1) logn > -dogt *- (n-1) dog var > loge lng (2) (5-1)> Tog ) mns 41 + 1 + 1 as ...ma s logl) tel + 2 [ log() J m ₃ oo as a 1 Thus m depends on E as well on a. Thus the sequence on does nol uniformly to f'zo. - Bowed converge