Exploring the Relationship Between Lim Sup and Supremum in Sequences

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In summary, lim sup, short for limit superior, represents the largest limit point of a sequence or function. It is different from lim inf, which represents the smallest limit point. To calculate lim sup, the sequence or function is analyzed to determine its limit points, and the largest one is chosen. Lim sup is significant in mathematical analysis and can help understand the behavior of a sequence or function. It can also be infinite if the sequence or function has no upper bound.
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IniquiTrance
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Suppose there is some sequence [itex] \{x_n\}_{n\in\mathbb{N}}[/itex]. Say we have a set of all the limits of all possible subsequences, would the supremum of this set be the superior limit of [itex] \{x_n\}_{n\in\mathbb{N}}[/itex]? What about if this value turns out to be 5, but there is a member of the sequence that is equal to 500, but is not the limit of any subsequence. Can the superior limit be lower than the supremum of the sequence? Thanks!
 
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  • #2
limsup may be lower than sup. Example xn = 1 + 1/n, sup = 2, limsup = 1.
 
  • #3
Thanks, that was quite helpful.
 

FAQ: Exploring the Relationship Between Lim Sup and Supremum in Sequences

What is lim sup?

Lim sup, short for limit superior, is a mathematical concept that represents the largest limit point of a sequence or function. It is also known as the upper limit or least upper bound.

How is lim sup different from lim inf?

While lim sup represents the largest limit point, lim inf, or limit inferior, represents the smallest limit point. In other words, lim inf is the lower bound of a sequence or function.

How is lim sup calculated?

To calculate lim sup, the sequence or function is first analyzed to determine its limit points. Then, the largest limit point is chosen as the lim sup. In other words, lim sup is the supremum (least upper bound) of the set of limit points.

What is the significance of lim sup?

Lim sup is an important tool in mathematical analysis and is used in various theorems and proofs. It helps to understand the behavior of a sequence or function and can provide insights into the convergence or divergence of a series.

Can lim sup be infinite?

Yes, lim sup can be infinite. This occurs when the sequence or function has no upper bound, and therefore, the largest limit point is infinite. In this case, we say that the lim sup does not exist or is equal to positive infinity.

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