What are the best definitions for limsup and liminf in sequences?

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In summary, according to Rudin, limsup is the smallest number such that any greater number is an eventual upper bound of the sequence. If x is bigger than limsup, then x is an eventual upper bound of the sequence, contrary to what Rudin says if x is less than the sequence for infinitely many n.
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I am trying to learn about limsup and liminf for sequences in R. Walter rudin in his book defines them to be the sup and inf of the set of subsequential limits (possibly including + and - infinity). Wikipedia defines limsup to be the smallest number such that any greater number is an eventual upper bound of the sequence.A few questions:1)How does the wikipedia definition cope with infinity and - infinity?2) Which definition is best to work with?

3) In his book Rudin states the result that if x is bigger than limsup, then it is an eventual upper bound of sequence. Bearing in mind he uses the first definition, he goes for a contradiction and says that if x is less than the sequence for infinitely many n, there exists a subsequential limit y exceeding x, contary to x being an upper bound. How he comes up with this y is mystery to me. Wonderful book but sometimes leaves out 'obvious steps'.
 
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Poirot said:
I am trying to learn about limsup and liminf for sequences in R. Walter rudin in his book defines them to be the sup and inf of the set of subsequential limits (possibly including + and - infinity). Wikipedia defines limsup to be the smallest number such that any greater number is an eventual upper bound of the sequence.A few questions:1)How does the wikipedia definition cope with infinity and - infinity?2) Which definition is best to work with?

3) In his book Rudin states the result that if x is bigger than limsup, then it is an eventual upper bound of sequence. Bearing in mind he uses the first definition, he goes for a contradiction and says that if x is less than the sequence for infinitely many n, there exists a subsequential limit y exceeding x, contary to x being an upper bound. How he comes up with this y is mystery to me. Wonderful book but sometimes leaves out 'obvious steps'.

In my opinion Wikipedia definition is clear and not ambigous so that I suggest to adopt it... if the sequence is umbounded on the upper part then the upper limit simply doesn't exist and if the sequence is umbounded on the lower part then the lower limit simply doesn't exist...

Kind regards

$\chi$ $\sigma$
 

FAQ: What are the best definitions for limsup and liminf in sequences?

What is a limit superior and limit inferior?

Limit superior and limit inferior are concepts in mathematical analysis that describe the maximum and minimum values that a sequence or function can approach as the independent variable approaches some limit. They are also known as the "upper limit" and "lower limit" respectively.

What is the difference between limit superior and limit inferior?

The main difference between limit superior and limit inferior is that limit superior is the largest value that a sequence or function can approach, while limit inferior is the smallest value. In other words, limit superior is an upper bound and limit inferior is a lower bound for the values of a sequence or function.

How do you calculate limit superior and limit inferior?

To calculate limit superior, you take the supremum (or highest possible value) of all the values that the sequence or function approaches. To calculate limit inferior, you take the infimum (or lowest possible value) of all the values that the sequence or function approaches. In simpler terms, you find the largest and smallest values that the sequence or function can get arbitrarily close to.

What is the significance of limit superior and limit inferior?

Limit superior and limit inferior can help us understand the behavior of a sequence or function as it approaches a certain limit. They can also be used to prove the convergence or divergence of a sequence or function. In addition, they have applications in various fields such as calculus, real analysis, and number theory.

Can limit superior and limit inferior be equal?

Yes, it is possible for limit superior and limit inferior to be equal. This would occur when the sequence or function is convergent, meaning it approaches a single value as the independent variable approaches its limit. In this case, the limit superior and limit inferior would both be equal to the limit of the sequence or function.

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