Chapter 3 - CBU
Chapter 3 - CBU
Chapter 3 - CBU
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
3.2. LIMIT THEOREMS 45Theorem (3.2.3). Suppose lim(x n ) = x and lim(y n ) = y.(a) lim(x n + y n ) = lim(x n ) + lim(y n ) = x + y.Proof. [The ✏ 2 technique.]Since lim(x n ) = x, 9 K 1 2 N 3 8 n K 1 , |x n x| < ✏ 2 .Since lim(y n ) = y, 9 K 2 2 N 3 8 n K 2 , |y n y| < ✏ 2 .Let K = max{K 1 , K 2 }. Then n K =) n K 1 and n K 2 =)|(x n + y n ) (x + y)| = |(x n x) + (y n y)| apple|x n x| + |y n y| < ✏ 2 + ✏ 2 = ✏. ⇤(b) lim(x n y n ) = lim(x n ) lim(y n ) = x y.Proof. Similar to the above.⇤