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iv) х lim 1 Yn lim-X Yn lim- (In particular, all these limits exist.)
2.12 Theorem. Suppose that {Xn) and (yn} are real sequences and that a € R. If {xn} and {yn} are convergent, then i) lim (Xn + yn) = lim xn + lim yn. 11-00 n-00 11-00 ii) lim (axn) = u lim xn and iii) lim (Xn yn) = (lim Xn) lim yn). n-00 If, in addition, yn 70 and lim Yn 70, then iv) lim, X lim - Yn lim, Yn (In particular, all these limits exist.)
Proof. Suppose that X, → x and yn →y as n → . i) Let e > 0 and choose N E N such that n > N implies xn - xl < 8/2 and Tyn - yl < €/2. Thus n > N implies 1(xin + yn) – (8 + 9)] = kx x1 + lyn - y<+ =E. ii) It suffices to show that ax, -ax → 0 as n → 00. But x - x 0 as n— , hence by the Squeeze Theorem, a (x - x) → as n → iii) By Theorem 2.8, the sequence (xn) is bounded. Hence by the Squeeze Theorem the sequences (xnn - y)) and ((X. - x)y} both converge to 0. Since Xnx - xy = x(y- y) + ( x - 1)y, it follows from parti) that XnYn → xy as n → . A similar argument estab- lishes part iv) (see Exercise 2.2.4). Theorem 2.12 can be used to evaluate limits of sums, products, and quotients. Here is a typical example.