Limits and Supremum: Is It True?

In summary, the conversation discusses the convergence of a sequence ##a_n## in norm to a limit ##a##, and the existence of a set ##S## such that the supremum of the inner product of ##a_n## and elements of ##S## is finite. The question is posed whether it is true that the supremum of the inner product of ##a## and elements of ##S## is also finite. The conversation also mentions the use of Latex for typing formulas on the forum. There is a question about whether this is a homework-type problem and a request for the person to show their work before receiving hints.
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We have ##a_n## converges in norm to ##a## and a set ##S## such that for all ##n\ge 0##
$$\sup_{s\in S} <a_n,s><+\infty .$$ Is it true that ##\sup_{s\in S} <a,s><+\infty##
 
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You need to use two hashes (or two dollars) for Latex delimiters.
 
  • #4
Is this a homework-type of problem? There is a format for those and you must show work before we can give hints.
Suppose ##M \in R## is such that ##sup<a_n,s> \lt M##. Also, suppose ##\epsilon \gt 0## and ##m\in N## are such that ##<a_n,a> \lt \epsilon## ##\forall n\gt m##. What can you say then?
 

1. What is a limit?

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It represents the value that the function is approaching, rather than the actual value at that point.

2. How is a limit calculated?

A limit is calculated by evaluating the function at values that are increasingly closer to the desired input value. This process is known as taking the limit. If the function approaches a specific value as the input gets closer and closer, then that value is the limit.

3. What is a supremum?

A supremum, or least upper bound, is the smallest possible number that is greater than or equal to all the numbers in a set. In other words, it is the smallest upper bound of a set of numbers.

4. How is the supremum of a set determined?

The supremum of a set is determined by finding the largest number in the set. If there is no largest number, then the supremum does not exist. In some cases, the supremum can be infinite.

5. What is the relationship between limits and supremum?

Limits and supremum are related in that the limit of a function can be equal to the supremum of a set. This occurs when the function is continuous and the supremum of the set is within the function's range. However, in most cases, the limit and supremum are two distinct concepts with different ways of being calculated.

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