1549901369-Elements_of_Real_Analysis__Denlinger_

2.5 Monotone Sequences 91

Theorem 2.5.3 (Monotone Con.vergence Theorem) Every bounded mono-lflf.WI

tone sequence converges. More precisely,

(a) if {an} is a monotone increasing sequence that is bounded above, then

n-HX) lim an= sup{ an: n EN};

(b) if {an} is a monotone decreasing sequence that is bounded below, then

lim an= inf{an: n EN}.

n-+oo

Proof. (a) Suppose {an} is bounded and monotone increasing. Since {an}

is bounded, the set {an : n E N} has an upper bound. By the completeness of

JR, 3u = sup{an: n EN}.

Let c > 0. By Theorem 1.6.6 (s-criterion for supremum), 3 no E N 3

an 0 > u - E:. But {an} is monotone increasing; therefore, n ~no::::} an~ ano·

Thus,

n ~ no ::::} an ~ an 0 > u - E:.

But since u =sup{ an: n EN},

'Vn E N, an :S u.

Putting together (2) and (3), we have

n ~ no ::::} u - c < an :S u < u + c

::::} u - E: < an < u + E:

::::} Ian - ul < E:.

Therefore, lim an = u = sup{ an : n E N}.
n-+oo
(b) Exercise 3. •

(2)

(3)

Corollary 2.5.4 (Fundamental Theorem of Monotone Sequences) A

monotone sequence converges if and only if it is bounded.

Proof. Exercise 4. •

*APPLICATION:

DECIMAL REPRESENTATION OF REAL NUMBERS

In Chapter 1 we defined the "Real Number System." However, we did
not say anything about the most familiar representation of real numbers-as
decimals. We are now in a position to make the connection, clearly and precisely.
A decimal expansion is an expression of the form
D = ± K.d1d2d3 · · · dndn+l · · · (4)

where K is a nonnegative integer and 'Vn E N, the "digit" dn is an element of
the set {O, 1, 2, 3, · · · , 9}. But what do we actually mean by the expression (4)?