136 Chapter 2 • Sequences
- For each of the following, prove or find a counterexample for which the
given equation is not true:
(a) lim (xn+ Yn) = lim Xn + lim Yn
(b)(c)n--+oo n-+oo n--+oon-+oo lim (XnYn) = ( n--+oo lim Xn) ( n-+oo lim Yn)
lim (xn+ Yn) = lim Xn + lim Yn
n->oo n->oo n->oo(d) lim (XnYn) = ( lim Xn) ( lim Yn)
n--+oo n--+oo n--+oo- Prove that
(a) If {x 11 } and {Yn} are bounded above, then lim (xn+ Yn):::; lim Xn+
n--+oo n-+oo
n->oo lim Yn·
(b) If {xn} is bounded above and r :::'.: 0, then lim rxn = r lim Xn.
n--+oo n--+oo
( c) If { Xn} and {Yn} are bounded sequences of nonnegative numbers,
then n-+oo lim (XnYn) ::=; ( n--+oo lim Xn) ( n-+oo lim Yn).
(d) Upper limits preserve inequalities. That is, if \in EN, Xn :::; Yn, then
lim Xn ::=; lim Yn·
n--+oo n-+oo- State and prove results similar to those of Exercise 2.9.6 for lower limits.
- Given any sequence {xn}, prove that lim (-xn)
n->oo- lim Xn and
n->oo
lim ( - x 11 ) = - lim Xn·
n->oo n->oo
- lim Xn and
- Suppose {xn} and {yn} are sequences of nonnegative numbers such that
Xn -t x f 0 and lim Yn = y. Prove that lim XnYn = xy. [Hint: Use
n-+oo n--+oo
subsequences and Theorem 2.9.10.]