
MHB Apprentice
#1
September 11th, 2019,
21:42
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.

September 11th, 2019 21:42
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MHB Oldtimer
#2
September 12th, 2019,
04:18
Originally Posted by
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.
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 $B$.