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Re: Question about sequences
Posted:
Feb 2, 2011 11:46 PM
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In article
<ac6c6398-d4c3-4fa6-90f7-f756500b065d@24g2000yqa.googlegroups.com>,
khosaa <qindars@gmail.com> wrote:
> Hi,
>
> 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"
He goes on to say "i.e. the sequence has an infinity of terms below -G."
> 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.
You would be correct. He is trying to say what it means to say -oo is a
limit point of a sequence (since " -oo " really isn't _anything_).
Subsequently, he says that if the sequence is bounded above but not
below and has no limit point other that -oo (i.e. no limit point at all
in the usual sense) then, by definition, the sequence "converges to
-oo". That is equivalent to "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).
He is defining the phrase " -oo is a limit point of a sequence" as a
whole. He is not giving a property of the (non-existent) thing -oo nor
is he saying that -oo is a limit point in the sense that he has
previously defined limit point. In the shaded box at the end of the
section he emphasizes that oo is not a real number.
"Limit point" is used two different ways: It is a property of a real
number relative to a set of real numbers and it is used to describe a
property of a sequence of real numbers. The two are only intuitively
related.
--
Paul Sperry
Columbia, SC (USA)
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