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Question about sequences
Posted:
Feb 2, 2011 9:40 PM


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"
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 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).
Thank you for any insights, Fran



