1549901369-Elements_of_Real_Analysis__Denlinger_

100 Chapter 2 • Sequences

1

Then neither an nor u+-is a lower bound for A, so 3an+l EA 3

n

u < an+l < min{an,u+ -

1

-}.

n+l

Thus, \In E N, an+i < an. So, {an} is a strictly decreasing sequence

1

of elements of A. Since u < an < u + - , we conclude by the first

n

squeeze principle that an __, u.

(b) Exercise 18. •

UNBOUNDED MONOTONE SEQUENCES

Theorem 2.5.14 (Unbounded Monotone Sequences Diverge to Infin-

ity)

(a) If {xn} is a monotone increasing sequence that is unbounded above, then

lim Xn = + oo.

n-+oo

(b) If { Xn} is a monotone decreasing sequence that is unbounded below, then

lim Xn = -oo.

n-+oo

Proof. (a) Suppose {xn} is monotone increasing and unbounded above.

Let M > 0. Since {xn} is unbounded above, M cannot be an upper bound for

{xn}, so :lno EN 3 Xn 0 > M. Since {xn} is monotone increasing, n 2: no =>

Xn 2: Xn 0 > M. Therefore, by Definition 2.4.1, n-+oo lim Xn = +oo.

(b) Exercise 19. •

Corollary 2.5.15 (a) A monotone increasing sequence either converges to a

real number or diverges to +oo.

(b) A monotone decreasing sequence either converges to a real number or

diverges to - oo.

Proof. These are immediate consequences of Theorem 2.5.14. •

A SLOWLY DIVERGENT SEQUENCE

(WILL FOOL A COMPUTER OR CALCULATOR)

With the easy accessibility of computers and calculators, students in el-
ementary calculus courses are being encouraged to explore sequences numeri-
cally, and "discover" by computation whether a sequence converges and to what