What is the definition of lim sup and how is it related to subsequential limits?

In summary, the formula \lim \text{sup } s_n = \text{sup } E defines the limit superior as the supremum of the set of all subsequential limits of a sequence $s_n$ in the real extended line. There are multiple equivalent ways to define lim sup, and it is not necessary to prove the definition. Examples can be found on a separate page.
  • #1
alyafey22
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Let $s_n$ be a sequence of real numbers and define $E$ to be the set of all subsequential limits of $s_n$ in the real extended line. Then we define the following

\(\displaystyle \lim \text{sup } s_n = \text{sup } E\)​

For some reason I don't quite understand the above formula , do we need to prove it ? It would be nice if you give some examples.
 
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  • #2
ZaidAlyafey said:
Let $s_n$ be a sequence of real numbers and define $E$ to be the set of all subsequential limits of $s_n$ in the real extended line. Then we define the following
\(\displaystyle \lim \text{sup } s_n = \text{sup } E\)​
For some reason I don't quite understand the above formula , do we need to prove it ? It would be nice if you give some examples.

We really don't prove a definition. There are many equivalent ways to define lim sup.
But there is no point in reinventing the wheel. Have a look at this page.
 
Last edited:

What is the upper limit of a sequence?

The upper limit of a sequence, also known as the supremum, is the largest value that the terms in a sequence can approach or converge to.

How is the upper limit of a sequence determined?

The upper limit of a sequence can be determined by taking the limit of the sequence as n approaches infinity. If the limit exists, it is the upper limit. If the limit does not exist, the upper limit is undefined.

Is the upper limit of a sequence always a part of the sequence?

No, the upper limit of a sequence may or may not be a part of the sequence. It only represents the maximum possible value that the terms in the sequence can approach.

Can the upper limit of a sequence be negative?

Yes, the upper limit of a sequence can be negative, zero, or positive. It depends on the values and behavior of the terms in the sequence.

What is the difference between the upper limit and the maximum value of a sequence?

The upper limit is the maximum possible value that the terms in a sequence can approach, while the maximum value is the largest value that the terms in the sequence actually attain. The maximum value may or may not be equal to the upper limit.

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