Chapter 3 - CBU

3.2. LIMIT THEOREMS 51Theorem (3.2.10). Suppose lim(x n ) = x and x nlim( p x n ) = p x.Proof. [Using the conjugate].First, x0 by Theorem 3.2.4. Let ✏ > 0 be given.Case x = 0 9 K 2 N 3 8 n K, |x n 0| < ✏ 2 ()0 apple x n < ✏ 2 () 0 apple p x n < ✏ () |x n 0| < ✏.0 8 n 2 N. ThenCase x > 0 Then p x > 0. 9 K 2 N 3 8 n K, |x n x| < p x✏ =)| p x np x| =( p x np x) · (pxn + p x)pxn + p x=|x n x|pxn + p x apple |x n x|p < xp x✏p x= ✏.HomeworkPages 69-70 #1d, 5b, 6bd (both find and prove)⇣ xn⌘Extra: If lim = x 6= 0, then (x n ) is not bounded.n⇣ xn⌘ ⌘bounded =) lim = 0 .n⇤⇣Hint: Prove (x n )