Infimum and Supremum of a Set (Need Help Finding Them)

[AutGuy98]

Hey guys,

I have this Intermediate Analysis problem that I need help finding the answer to. This is what the question asks:

"Find the supremum and infimum of each of the following sets (considered as subsets of the real numbers). If a supremum or infimum doesn’t exist, then say so. No formal proof is necessary, but give a brief justification."

This is the set in question: B={(-1)^n+((-1)^n+1)/(2n)): n is a subset of Z (the set of integers) - {0}} (meaning "not including 0).

I started out by plugging in values for n from -5 to 5, not including 0, to see the answers produced, but I wasn't able to identify a pattern between any of them or anything like that. Not sure where to go from here with the problem, so any help you could give me would be helpful without a doubt. Thanks in advance.

 

[Opalg]

Hey guys,

I have this Intermediate Analysis problem that I need help finding the answer to. This is what the question asks:

"Find the supremum and infimum of each of the following sets (considered as subsets of the real numbers). If a supremum or infimum doesn’t exist, then say so. No formal proof is necessary, but give a brief justification."

This is the set in question: B={(-1)^n+((-1)^n+1)/(2n)): n is a subset of Z (the set of integers) - {0}} (meaning "not including 0).

I started out by plugging in values for n from -5 to 5, not including 0, to see the answers produced, but I wasn't able to identify a pattern between any of them or anything like that. Not sure where to go from here with the problem, so any help you could give me would be helpful without a doubt. Thanks in advance.

Hi AutGuy, and welcome to MHB.

Let $x_n = (-1)^n + \dfrac{(-1)^{n+1}}{2n} = (-1)^n\left(1 - \dfrac1{2n}\right)$. Then $|x_n| = 1 - \dfrac1{2n}$.

If $n$ is positive then $|x_n|<1$ and if $n$ is negative then $|x_n|>1$. Also, if $n$ is small and negative then $|x_n|$ will be larger than if $n$ is large and negative.

In calculating $x_n$ for n from -5 to 5, you found (I hope) that the greatest and least values of $x_n$ occurred when $n=-2$ and $n=-1$.

From those hints, you should be able to "give a brief justification" of the fact that theose values are the sup and inf of the set $B$.

 

[soloenergy]

Hi there,

It looks like you're trying to find the supremum and infimum of the set B. To do this, we need to first understand what these terms mean.

The supremum of a set is the smallest number that is greater than or equal to all the numbers in the set. In other words, it is the "least upper bound" of the set. The infimum, on the other hand, is the largest number that is less than or equal to all the numbers in the set. It is the "greatest lower bound" of the set.

Now, let's look at the set B. It is a set of numbers that are generated by plugging in different values for n. We can see that the terms inside the parentheses alternate between -1 and 1, and the terms outside the parentheses are all positive. When n is a positive even number, the term inside the parentheses is 1, and when n is a negative even number, the term is -1. Similarly, when n is a positive odd number, the term inside the parentheses is -1, and when n is a negative odd number, the term is 1.

Therefore, we can see that the terms in this set are always either 1 or -1, and they are alternating between these two values. This means that the supremum of the set would be 1, as it is the smallest number that is greater than or equal to all the numbers in the set. Similarly, the infimum would be -1, as it is the largest number that is less than or equal to all the numbers in the set.

To summarize, the supremum of the set B is 1, and the infimum is -1.

I hope this helps! Let me know if you have any other questions.