Module 1
:
Real Numbers, Functions and Sequences
Lectur
e
3
:
Monotone Sequence and Limit theorem [ Section 3.3 : Some Extensions of the Limit Concept ]
Practice Exercises 3.3 : Extension of Limit Concept
(1)
Let
and
be two sequences of positive real numbers such that
exists and
Show that if
is convergent to
, then
.
(2)
Give an example to show that conclusion of (1) need not hold for the cases when
(3)
If
, show that
.
(4)
Let
be an unbounded sequence. Show that there exists a subsequence of
which is convergent to
or
.
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and 
be two sequences of positive real numbers such that 
exists and 
Show that if 
.
, show that 
.
