Sufficient conditions for a = lim inf xn

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In summary, the theorem in "Elementary Classical Analysis" by Marsden and Hoffman is left unproved. It states that for every e>0 there is an n such that xn<a+e. For all e>0 and all M, there is an n>M with xn<a+e. If xn is bounded below, lim inf xn is the infimum of the set of all cluster points of xn. Setting e=(a-b)/2 in (i) gives (a+b)/2 <xn for all n>=N. This means that for every n>=N, there is an n>M with xn<a+e and xn is
  • #1
samkolb
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This is part of a theorem which is left unproved in "Elementary Classical Analysis" by Marsden and Hoffman.

Let xn be a sequence in R which is bounded below. Let a be in R.

Suppose:

(i) For all e > 0 there is an N such that a - e < xn for all n >= N.

(ii) For all e > 0 and all M, there is an n > M with xn < a + e.

Show that a = lim inf xn.

(Definition: When xn is bounded below, lim inf xn is the infimum of the set of all cluster points of xn. If xn has no cluster points, then lim inf xn = + infinity. If xn is not bounded below, then lim inf xn = - infinity.)

I was able to use (i) and (ii) to show that a is the limit of a subsequence of xn, hence a is a cluster point. So to show that a = lim inf xn, it is sufficient to show that a is a lower bound for the set of cluster points. This is what I can't do. Any suggestions?
 
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  • #2
Well, suppose b < a is a cluster point. Doesn't that give you a problem with (i).?
 
  • #3
So I set e = a - b in (i) and get b < xn for all n>=N. I don't see the problem with this.

I think this implies all cluster points x satisfy b <= x, which means b <= a, but now I'm back where I started.
 
  • #4
Try e = (a - b)/2. Can't you show all but finitely many of the xn are bounded away from b?
 
  • #5
I think I get it now.

Setting e = (a -b)/2 in (i) gives (a + b)/2 < xn for all n >= N

==> (a - b)/2 < xn - b for all n >= N

==> no subsequence of xn can converge to b, since (a - b)/2 > 0

==> b is not a cluster point.


thanks!
 

FAQ: Sufficient conditions for a = lim inf xn

1. What does "sufficient conditions for a = lim inf xn" mean?

This phrase refers to a concept in mathematical analysis known as the limit inferior. It states that the limit inferior of a sequence xn is equal to a if and only if certain conditions are met.

2. What are these conditions that must be met for a = lim inf xn?

The conditions are that the sequence xn must be bounded below, monotone, and convergent. Additionally, the limit of xn must be equal to a.

3. How is the limit inferior calculated?

The limit inferior is calculated by finding the infimum (greatest lower bound) of all subsequential limits of the sequence xn. In other words, it is the smallest value that the sequence approaches as n approaches infinity.

4. What is the significance of the limit inferior?

The limit inferior provides a lower bound for the sequence xn, meaning that all values of xn must be greater than or equal to this limit. It also helps to determine the convergence or divergence of a sequence.

5. Can the limit inferior be equal to a value other than a?

No, the limit inferior can only be equal to a if all the conditions are met. If any of the conditions are not satisfied, then the limit inferior will be a different value.

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