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

AutGuy98

New member
Sep 11, 2019
20
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

MHB Oldtimer
Staff member
Feb 7, 2012
2,679
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 $B$.