Question

JUST PART C

10. Let X denote the subset of R consisting of all sequences (x1,x2,...) such that x converges. (You may assume the standard facts about infinite series. In case they are not familiar to you, we shall give them in Exercise 11 of the next section.) .(6,X) = (3x) (a) Show that if x, y e X, then xyil converges. [Hint: Use (b) of Exercise 9 nu er og to show that the partial sums are bounded.] 2uur (b) Letc. e R. Show that if x,y e X, then so are x +y and cx. (c) Show that Vo i t dou2 X 290nup32 B 10 1/2 d(x, y) = - 902 is a well-defined metric on X.

Answer 1

O Since Eur converges » Exi co Now, First of all, we prove that d is well defined le. dlx,y)<a (-492 = ((ud+ (4.3)? - ((x + y ) ²)" = [owe + 14 ) } ] ² Therefore < x to > day) <0 d is well defined.

a foren diming = [ (x - y)? ] (1) Since, (ni-yu)? o - lui-yah2 so r i=1,2-. ti - [days so it ny ex (ii) Let dix,y) = 0 W U V Ž (x - you o ni-ye = 0 ni = Ye & & 1=1,2- i=1,2... u Conversely, let x=y di = Yi & 1=1,2-- (xi-Yc)² =0 &i=1,2- Elxi-yu)2 = 0

= 0 - » dix,y) we have [dlx,y) zo if and only if x=4 BLUE TORONTO diuig - [ 6YouTube = [(y-487² E (any)? - 14:07] = dlyn) Idency) = dry,m) , - r ny EX (iv) To prove the triangle inequality, we use Minkowski inequality & lace wiggle = ( 16.0)² + ( 84.30'e Nowy dlny) = ( (ve-you) < [3(x+7) + (z- y)

Li=1 List dix,y) s dlxiz) + diz,y) So, from above is four cas conditions, d is well defined metric on x.