1549901369-Elements_of_Real_Analysis__Denlinger_

66 Chapter 2 • Sequences

lfiJVM Theorem 2.2.13 (Algebra of Limits) Suppose {xn} and {yn} are conver-
,,, ,,, gent sequences and c E R
Then
(a) n-+oo lim (cxn) = c · n-+oo lim Xni

(b) n-+oo lim (xn + Yn) = n-+oo lim Xn+ n-+oo lim Yni

(c) n-+oo lim (xn - Yn) = n-+oo lim Xn- n-+oo lim Yni

(d) n-+oo lim (XnYn) = n-+oo lim Xn· n-+oo lim Yni

(e) lim (2-) = -

1

• n->oo Yn lim Yn
  n->oo

(if lim Yn =!= 0);

n->oo

(f) l" ( Xn) J!_.~ Xn

n.:.,~ Yn = lim Yn

n->oo

(if lim Yn=t'=O);

n->oo

(g) lim (y!Xn") =. f lim Xn (if lim Xn 2: 0, and 3 n1 3 n 2: n1 =? Xn 2: 0).

n-+oo V n-+oo n-+oo

fi~ o::J.~

Proof. Suppose { Xn} and {yn} are convergent sequences, and c E R In

fact, suppose Xn---+ L and Yn---+ M. Then

(a) Case 1 (c =!= 0) : Let c; > 0. Since Xn ---+ L , 3no E N 3 n 2: n 0 =?

€

lxn - LI< i;;r·

Then, n 2: no

€

=? lcxn - cLI = lei lxn - LI < lei · i;;f

=? lcxn - cLI < c:.

Therefore, cxn ---+ cL.

Case 2 (c = 0): Exercise 9.

(b) Let c; > 0.

Since Xn---+ L , 3n1 EN 3 n 2: n1 =? lxn - LI<;.

Since Yn---+ M, 3n 2 EN 3 n 2: n2 =? IYn - Ml < ;.