Module 1
:
Real Numbers, Functions and Sequences
Lecture
3
:
Monotone Sequence and Limit theorem [ Section 3.2 : Limit Theorems on Sequences ]
Let
.
Then,
by exercise (ii). Suppose both x, y
0. Let an
be given Choose
such that
and
Note that
Then,
Hence,
. This proves (ii) when
The case when
is easy and is left as an exercise.
(iii)
To prove (iii), we have to show that given an
and such that
Since
and
,we have
such that
.
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_clip_image002_0001.gif){kind=small}
. 
by exercise (ii). Suppose both x, y
0. Let an
be given Choose 
such that 
and 
_clip_image002_0002.gif){kind=small}
Then, 
. This proves (ii) when _clip_image002.gif){kind=small}
The case when _clip_image002_0000.gif){kind=small}
is easy and is left as an exercise. 
and such that 
and 
,we have 
such that_clip_image002_0002.gif)
.
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