Question

4. A function sequence (fn)neN İs unifonnly bounded on D if there is an M0 such that If, (xl s M for allxin D and all in N. Show that if s uniformly convergent on D and each fn is bounded on D, then (fn)neN 1S uniformly bounded on D. Use this to conclude that the function sequence in Example 8.3 is not uniformly convergent. Example 8.3 Let f : (0.1)-R be given by falx)-nx(I-2for eaclh n in (Higure 8.2) 14- 8 0.8 n-3 0.6 0.4 0.2 0.2 0.4 0.6 0.8 Figure 8.2

Answer 1

Le1 彳S-3 is uniso3 my converges to f Ahat E IR Such thas(Mn Vi o Nou Fsom since each n ÍS bounded so No w M = max uniformiy bounded, on

」n: [0,1] →IR be given by m (a)o ラ.of n (1-12)n-lFa? (1+ 2m) +1) Eto,1] o l+ 2M ) att ans its maximum vcuuu au 2ach inta) is bounded on (o.1] 2 an 2+%, is is no nisosmay bound on (o,j lence, n 2 is not e