Module 1
:
Real Numbers, Functions and Sequences
Lectur
e
3
:
Monotone Sequence and Limit theorem [ Section 3.3 : Some Extensions of the Limit Concept ]
3.3.5
Theorem:
Let
be a sequence. Then
is convergent to
iff every subsequence of
is convergent to
Proof
Clearly, if
is covergent to
, so is its every subsequence.
Conversely, suppose that every subsequence is convergent to
.
Suppose,
is not convergent to
.
Then
such that
such that
.
We start with
, and choose
and then choose
, and in general choose
with
.
Then,
will be a subsequence of
not convergent to
, a contradiction.
Hence,
.
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iff every subsequence of 
, so is its every subsequence. 
such that 
such that 
. 
, and choose 
and then choose 
, and in general choose 
with 
. 
will be a subsequence of 
not convergent to 
. 
