Royden Review:LimSup, LimInf are the Largest/Smallest Limit Points of {a_n}

• Bacle
In summary, the conversation discusses the sequence {a_n}, and the goal of showing that LimSup and LimInf are the largest and smallest limit points of the sequence. The speaker mentions that if the sequence converges, then LimSup and LimInf are equal and if the limit is infinity, then infinity is the limit point. They also provide a proof for LimSup and LimInf being limit points in the case that the sequence does not converge. However, the speaker is stuck trying to show that LimSup is the largest limit point, but believes that once this is proven, it will automatically show that LimInf is the smallest limit point. The conversation ends with a discussion about the definitions of LimSup and LimInf and the fact that a
Bacle
Hi, All:

Let {a_n}; n=1,2,... be sequence. I am trying to show LimSup and LimInf are the largest and smallest limit points of {a_n}. This is what I got so far:

i) If {a_n} converges, to, say, a<oo, then LimSup=LimInf, and we're done, since we have a unique limit point L. If a=oo, then oo is the limit point.

I think I can show ( here in ii) below ) that Lim Sup, Lim Inf are both limit points of {a_n}, but I cannot show they are the largest, smallest respectively.

LimSup is a limit point of {a_n} if {a_n} does not converge:
Proof:
ii)If {a_n} does not converge, then it is not strictly monotone, so we can extract monotone

non-increasing and monotone non-decreasing subsequences ( by using, e.g., the lim sup

and lim ii) lim sup, lim inf are both limit points; in the case of lim sup, the sequence: a_n' :

{ sup_k>n(a_n)} is monotone non-increasing; by the LUB property,L= LimSup{a_n} :=inf_n

(a_n') exists, and it is a limit point of {a_n}; by contradiction, if L were not a limit point of

{a_n}, there would be e>0 with L> a_k-e for all a_k. But then L is not the lub of the

monotone-decreasing subsequence; a_k-e is the LUB.

I am stuck trying to show that Lim Sup is the largest limit point; I am sure the proof that

Lim Inf is the smallest limit point is automatic after knowing this one.

Thanks.

What is your definition of "limsup" and "liminf". The definition I learned was "[/itex]a= limsup a_n[/itex] if and only if a is the supremum of the set of all subsequential limits of $a_n$" and "$b= liminf a_n$ if and only if b is the infimum of the set of all subsequential limits of $a_n$".

But notice that I used "sup" and "inf", NOT "largest" and "smallest". A sequence may not have "largest and smallest limit points".

Let's say that L is a limit point of the sequence. Can you prove that

$$\sup_{n\geq k}~{a_n}\geq L$$

for every k?

1. What is the significance of LimSup and LimInf in the Royden Review?

LimSup and LimInf, short for limit supremum and limit infimum, are two important concepts in real analysis. In the context of the Royden Review, they refer to the largest and smallest limit points of a sequence {a_n}, respectively. These values help us understand the behavior of a sequence and its convergence or divergence.

2. How do you calculate LimSup and LimInf?

To calculate LimSup, you first need to find the set of all the limit points of the sequence {a_n}. Then, take the supremum (or the least upper bound) of this set. Similarly, to calculate LimInf, you need to find the set of all limit points and take the infimum (or the greatest lower bound) of the set. In simpler terms, LimSup is the largest value that the sequence can get arbitrarily close to, and LimInf is the smallest value that the sequence can get arbitrarily close to.

3. Why are LimSup and LimInf important in real analysis?

LimSup and LimInf help us understand the behavior of a sequence in terms of its convergence or divergence. If LimSup and LimInf are equal, the sequence is said to converge. If they are not equal, the sequence is said to diverge. Additionally, these values also help us find the limit of a sequence, if it exists.

4. Can LimSup and LimInf be infinite?

Yes, LimSup and LimInf can be infinite. In fact, they can take on any real value, including infinity and negative infinity. This can happen when the sequence {a_n} is unbounded or does not have a limit.

5. How are LimSup and LimInf related to the terms "supremum" and "infimum"?

The terms "supremum" and "infimum" are used in set theory to refer to the least upper bound and greatest lower bound of a set, respectively. In the context of the Royden Review, these terms are used to calculate LimSup and LimInf, which are the supremum and infimum, respectively, of the set of limit points of a sequence {a_n}.

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