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Question: show that rp is complete hint use the method in...

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Show that RP is complete. (Hint: Use the method in Example 5.2.2(b) to show that a sequence in RP is a Cauchy sequence if and only if each of the sequences of its coordinates is a Cauchy sequence in R.) Show that a sequence {(x1,x3) in R#x Rq (see Proposition 5.1.7) converges to (xi, x2) if and only if the same thing happens when we consider the sequence as belonging to RP+g

so we cannot use the definition of convergence we used in IR. Instead we use the generalization of the equivalent formulation from Proposition 1.3.3. 5.2.1. Definition. A sequence {x,.) in (X, d) converges to x if for every € > 0 there is an integer N such that d(X, Xn) < e when n N. The notation for this is → x or x = lim, xn. 5.2.2. Example. (a) A sequence in IR converges in the sense of $1.3 if and only if it converges in R considered as a metric space. (This is the content of (1.3.3).) (b) If xn = (x1, . . . ,xh) and (xi, . . . ,x®) are in Rp, then xn → x if and only if d → x/ in R for l < j p. In fact if xn → x, then we need only note that for l j p. Irn-刈: d(An, x) to conclude that x-x, Now assume that x-x! for l j p. Lete > 0 and for l < j p choose Nj such that lxn-dl < €/ when n > Ni. If we put N = max {M, . . . , Npl and n N, then J. Now assume that → → x in RP. Hence d(Ana) < E for n > N and we have that x sk. イ

We close with the following, Whose I0r1 LI 5.1.7. Proposition. If (X, d) and (Z, p) are two metric spaces and we define the function 8 : (X x z) x (X x Z) R by then δ is a metric on X x Z.

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