Solving Supremum Question: Is 4 the Right Answer?

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In summary, the conversation is discussing a question about finding the supremum of a set and the attempt at solving it using l'hopital's rule. However, the answer provided did not fully prove that 4 is the supremum and was missing some important elements. The person asking the question wonders if they deserve any points for their attempt.
  • #1
Cankur
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Hello!

I had a test in which the question that I will present here was asked. I got no points for my attempt at a solution. Do you think that I was still on the right track and that I deserve partial points? Here is the question:

"A number M is said to be an upper bound to a set A if M [itex]\geq[/itex] x for every x[itex]\in[/itex] A. A number S is said to be supremum of a set A if S is the smallest upper bound to A.

Assume that:

A = {(4n2)/(n2+1) : n [itex]\geq[/itex]0 is an integer}.

Show that supremum of A is 4."

And here is what I wrote as an answer (not verbatim, but translated from another language):

"Since n does not have an upper limit, it can go toward infinity. In this case:

A = lim (n [itex]\rightarrow[/itex] [itex]\infty[/itex]) (4n2)/(n2+1)=[itex]\infty[/itex]/[itex]\infty[/itex]. This shows that we can use l'hopital's rule. After using l'hopital's rule twice we get that A = 4. In other words, this gives us supremum. Since n always can be even bigger, this is just the smallest upper bound.

Answer: By using l'hopital's rule twice, I have shown that supremum A is 4."

Out of the possible 4 points that one could get on that question, I got 0. Was it justified?

Thanks in advance!
 
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  • #2
Essentially, what you showed was that the limit, as n goes to infinity, of that sequence is 4. That does NOT prove that 4 is the supremum. For example, the limit of 1/n, as n goes to infinity is 0 but 0 is definitely not the supremum!

Here, you would also have to show that your sequence is increasing and you did not do that.

Oh, and I certainly would not have used L'Hopital's rule for that limit: just divide both numerator and denominator by n2.
 
  • #3
In simple terms: (4n2)/(n2+1)=4/(1+1/n2) < 4.

As n becomes infinite limit is 4. Another way is assume the sup = a < 4, then you can find a large enough n so the expression > a: contradiction.
 
  • #4
Thanks for the answers! I see that there were quite essential things that I missed. But you don't think I deserve some points for my answer?
 

FAQ: Solving Supremum Question: Is 4 the Right Answer?

What is the Supremum Question and why is it important?

The Supremum Question is a mathematical concept that asks for the smallest upper bound of a set of numbers. It is important because it helps to determine the maximum value of a given set, which is useful in many real-world applications.

What is the process for solving the Supremum Question?

The process for solving the Supremum Question involves finding the upper bound of the set of numbers and then determining if it is the smallest upper bound. This can be done through various mathematical methods such as using inequalities or the definition of supremum.

Why is 4 often considered to be the right answer for the Supremum Question?

4 is often considered to be the right answer for the Supremum Question because it is the smallest upper bound for many common sets of numbers. For example, the set of natural numbers has a supremum of 4, as there is no number larger than 4 that is also a natural number.

Are there any cases where 4 may not be the right answer for the Supremum Question?

Yes, there are cases where 4 may not be the right answer for the Supremum Question. This can occur when the set of numbers has a larger upper bound, such as in the set of real numbers where the supremum is infinity. It can also occur when there is no upper bound, such as in the set of all positive real numbers.

How is the Supremum Question used in real-world applications?

The Supremum Question is used in various real-world applications, such as in economics and finance to determine the maximum price of a product or the maximum value of a stock. It is also used in computer science to optimize algorithms and in engineering to design systems with maximum efficiency.

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