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Topic: Question about sequences
Replies: 3   Last Post: Feb 4, 2011 2:19 AM

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Posts: 45
Registered: 1/12/07
Question about sequences
Posted: Feb 2, 2011 9:40 PM
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I am studying from Malik's book "Mathematical Analysis", and I came
across a definition that seems strange to me. I want to check and see
if I am not missing something.
In his section on "non-convergent sequences" Malik defines a "sequence
unbounded on the left" as follows:

"If the sequence {S_n} is unbounded on the left then we say that -
infinity is a limit point of the sequence, and to each positive number
G, however large, there corresponds a positive integer m such that S_n
< -G for every n >=m"

I think this does not sound right since it for any given negative
number, it requires *all* the sequence members past some index to lie
below said negative number. Hence, by Malik's definition, the sequence
S_n = n * (-1)^n, would not be unbounded on the left. I would think
that a better definition would have been that for every positive
number G, there exists a natural number m such that S_m < -G.

It also seems somewhat strange to talk about infinity (or negative
infinity) as a limit point of a sequence. I could see how -infinity
could be the lim inf of a sequence, but Malik's defn of limit point
of a sequence is that for every epsilon, an infinite number of terms
in the sequence are required to be in the open epsilon nbd of the
limit point. I would like to think he is making this definition for a
real-valued limit point and one would adopt a different definition for
a limit point in the extended reals (although he does not seem to do
this at any point in the text).

Thank you for any insights,

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