Understanding Superior Limits and Their Definition in Real Numbers - Explained

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In summary, the conversation discusses the definition of a superior limit of a sequence, which is the supremum of the set of real numbers that have a subsequence converging to it. The confusion lies in understanding the conditions a) and b), which state that for any small positive value, there are infinitely many terms in the sequence that are either less than or greater than the limit by that value. The speaker is unsure how these conditions are derived and if they are trying to prove something.
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radou
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So, my book says that the real number L is a superior limit of the sequence (an) iff the following holds:

a) [itex]\forall \epsilon > 0[/itex], [itex]a_{n} < L + \epsilon[/itex] holds, for almost all terms of the sequence,
b) [itex]\forall \epsilon > 0[/itex], [itex]L - \epsilon < a_{n}[/itex] holds, for an infinite number of terms of the sequence.

OK, the two facts confuse me. I know that "almost all terms of the sequence" means "all terms, except a finite number of terms". If L is a superior limit, then it is the supremum (by definition) of the set A of all real numbers [itex]a \in \textbf{R}[/itex] for which there exists a subsequence (bn) of the sequence (an) such that [itex]\lim_{n \rightarrow \infty} b_{n} = a[/itex]. Hence, for every [itex]\epsilon' > 0[/itex], there exists an element x of A such that [itex]L - \epsilon' < x[/itex]. Because x belongs to A, for some [itex]\epsilon'' > 0[/itex], the interval [itex]<x - \epsilon'', x+ \epsilon''>[/itex] contains almost all elements of a subsequence of the sequence (an). This is where I'm stuck and highly confused.
 
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  • #2
Are you confused by the concept, or are you trying to prove something?
 
  • #3
NateTG said:
Are you confused by the concept, or are you trying to prove something?

The concept (eg definition) is clear to me, I just don't understand how one arrives at a) and b).
 

1. What are superior limits in real numbers?

Superior limits in real numbers refer to the largest possible value that a sequence of real numbers can approach as the index approaches infinity. It is also known as the limit superior or the upper limit.

2. How do superior limits differ from regular limits?

Superior limits differ from regular limits in that they consider the behavior of a sequence of numbers as the index approaches infinity, rather than focusing on the behavior of a single point in the sequence.

3. What is the definition of superior limits?

The definition of superior limits is the supremum of all possible accumulation points of a sequence of real numbers. In other words, it is the largest number that a sequence can approach without ever reaching it.

4. How are superior limits useful in real-world applications?

Superior limits can be useful in real-world applications, such as in statistics and economics, where they can help determine the maximum possible value for a certain variable or the upper bound for a set of data.

5. Can superior limits be calculated for all sequences of real numbers?

Yes, superior limits can be calculated for all sequences of real numbers. However, some sequences may not have a defined superior limit, such as those that do not approach a single value as the index approaches infinity.

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