Text_Template

Module 1 : Real Numbers, Functions and Sequences

Lecture 3 : Monotone Sequence and Limit theorem [ Section 3.3 : Some Extensions of the Limit Concept ]

3.3.3	Theorem:
	Let be a sequence of real numbers.
(i)	If converges to , then it is not bounded above.
(ii)	If converges to , then it is not bounded below.
(iii)	If is monotonically increasing and not bounded above then, converges to .
(iv)	If is monotonically decreasing and not bounded below then, converges to .
	Proof
	Proofs of (i) and (ii) are obvious.
	To prove (iii), let be arbitrary. Since, is not bounded above, there exists some such that . Since is monotonically increasing, this implies that for every Hence, converges to .
	Proof of (iv) is similar. Back