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Re: Question about sequences
Posted:
Feb 2, 2011 11:46 PM


In article <ac6c6398d4c34fa690f7f756500b065d@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 "nonconvergent 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 > realvalued 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 (nonexistent) 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)



