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 .
3.3.4
Definition:
Let be a sequence and let be as strictly increasing sequence of natural numbers. Then is called a subsequence of . In some sense, is a part of with due regard to the order of the terms.
3.3.5
Theorem:
Letbe a sequence. Thenis convergent toiff every subsequence ofis convergent to
be as strictly increasing sequence of natural numbers. Then 
is called a subsequence of 
iff every subsequence of
