22. Prove that parts

(c) and

(d) of Exercise 6 have nothing to do with the base-10 assumption in the decimal expansion. In other words, if b is any positive integer, bb2, and each such x A ½0; 1 is expanded in base-b so that x ¼ 0:a1a2a3 . . . , where each aj A f0; 1; 2; . . . ; b  1g, then again:

(a) With x A Ry defined by x ¼ ðx1; x2; . . . ; xj ; . . .Þ, where xj ¼ aj b j , and xn A Ry0 is defined as before, we have that kx  xnkp ! 0 for all p, 1a pay.

(b) With y A Ry defined by y ¼ ða1; a2; . . . ; aj ; . . .Þ, where xj ¼ aj , we have that y A lp only for p ¼ y; yet even in this case ky  ynky n 0, where yn ¼ ða1; a2; . . . ; an; 0; 0;
0; . . .Þ unless y A Ry0 .