Editing Monotone convergence theorem (section)

==Convergence of a monotone sequence of real numbers==

Every bounded-above monotonically nondecreasing sequence of real numbers is convergent in the real numbers because the supremum exists and is a real number. The proposition does not apply to rational numbers because the supremum of a sequence of rational numbers may be irrational.

===Proposition===
(A) For a non-decreasing and bounded-above sequence of real numbers 
:<math>a_1 \le a_2 \le a_3 \le...\le K < \infty,</math> 
the limit <math>\lim_{n \to \infty} a_n</math> exists and equals its [[supremum]]:
:<math>\lim_{n \to \infty} a_n = \sup_n a_n \le K.</math>

(B) For a non-increasing and bounded-below sequence of real numbers 
:<math>a_1 \ge a_2 \ge a_3 \ge \cdots \ge L > -\infty,</math>
the limit <math> \lim_{n \to \infty} a_n</math> exists and equals its [[infimum]]:
:<math>\lim_{n \to \infty} a_n = \inf_n a_n \ge L</math>.

===Proof===
Let <math>\{ a_n \}_{n\in\mathbb{N}}</math> be the set of values of <math> (a_n)_{n\in\mathbb{N}} </math>. By assumption, <math>\{ a_n \}</math> is non-empty and bounded above by <math>K</math>. By the [[least-upper-bound property]] of real numbers, <math display="inline">c = \sup_n \{a_n\}</math> exists and <math> c \le K</math>. Now, for every <math>\varepsilon > 0</math>, there exists <math>N</math> such that <math>c\ge a_N > c - \varepsilon </math>, since otherwise <math>c - \varepsilon </math> is a strictly smaller upper bound of <math>\{ a_n \}</math>, contradicting the definition of the supremum <math>c</math>. Then since <math>(a_n)_{n\in\mathbb{N}}</math> is non decreasing, and <math>c</math> is an upper bound, for every <math>n > N</math>, we have 
:<math>|c - a_n| = c -a_n \leq c - a_N = |c -a_N|< \varepsilon. </math>
Hence, by definition <math> \lim_{n \to \infty} a_n = c =\sup_n a_n</math>.

The proof of the (B) part is analogous or follows from (A) by considering <math>\{-a_n\}_{n \in \N}</math>.

===Theorem===
If <math>(a_n)_{n\in\mathbb{N}}</math> is a monotone [[sequence]] of [[real number]]s, i.e., if <math>a_n \le a_{n+1}</math> for every <math>n \ge 1</math> or <math>a_n \ge a_{n+1}</math> for every <math>n \ge 1</math>, then this sequence has a finite limit [[if and only if]] the sequence is [[bounded sequence|bounded]].<ref>A generalisation of this theorem was given by {{cite journal |first=John |last=Bibby |year=1974 |title=Axiomatisations of the average and a further generalisation of monotonic sequences |journal=[[Glasgow Mathematical Journal]] |volume=15 |issue=1 |pages=63–65 |doi=10.1017/S0017089500002135 |doi-access=free }}</ref>

===Proof===
* "If"-direction: The proof follows directly from the proposition.
* "Only If"-direction: By [[(ε, δ)-definition of limit]], every sequence <math>(a_n)_{n\in\mathbb{N}}</math> with a finite limit <math>L</math> is necessarily bounded.