Supremum and Infimum of a Set Containing Rational Numbers

[Felafel]

Homework Statement

Hello. I think I've managed to solve the exercise, but I'd like it to be checked and I'd also like to kow whether it is well-written and explained enough.

The problem is: find the supremum and infimum of this set:
A= \{ \frac{m}{n}+4\frac{m}{n} : m,n \in \mathbb{N}^*\}

The Attempt at a Solution

if m=n then 5 is the only element of the set. Therefore it is its minimum and maximum and also its supremum and infimum.
if m<n the infimum (and also the minimum) is $\frac{17}{2}$ because the minimum values for m and n are 1 e 2.
if n<m the infimum is 4, for the same reason.
either when m<n or n<m the supremum is $+\infty$

 

[micromass]

I think you are severely misunderstanding the notation. The notation

A=\{\frac{m}{n}+4\frac{m}{n} | m,n\in \mathbb{N}^*\}

means that we take all m and n in \mathbb{N}^*

For example:

• 5 is an element of A since we can take m=n=1
• 5/2 is an element of A since we can take m=1 and n=2
• 10 is an element of A since we can take m=2 and n=1

These are three examples of elements in A. There are others.

We are letting m and n vary among the natural numbers. They can be truly anything.

 

[Felafel]

ok, should I put the three cases m>n,m<n,m=n together then, and call infimum and supremum the highest and lowest values I find in the union?

 

[micromass]

ok, should I put the three cases m>n,m<n,m=n together then, and call infimum and supremum the highest and lowest values I find in the union?

I don't really know why you are so set in separating the problem in cases m>n, m<n and n=m. That doesn't seem necessary here.

What elements of A do you get if m=1 and n varies?
What elements of A do you get if n=1 and m varies?
What does that tell you?

 

[Felafel]

I don't really know why you are so set in separating the problem in cases m>n, m<n and n=m. That doesn't seem necessary here.

What elements of A do you get if m=1 and n varies?
What elements of A do you get if n=1 and m varies?
What does that tell you?

hahaha don't know why. but what if the set were more complicated? like
$B= \{ \frac{mn}{4m^2+n^2} : m \in \mathbb{Z}, n \in \mathbb{N}^* \}$
wouldn't it be a good idea to separate the cases?

 

[micromass]

hahaha don't know why. but what if the set were more complicated? like
$B= \{ \frac{mn}{4m^2+n^2} : m \in \mathbb{Z}, n \in \mathbb{N}^* \}$
wouldn't it be a good idea to separate the cases?

It really depends on which sets you are given. Right now, I would be inclined to write

\frac{mn}{4m^2 + n^2} = 4\frac{n}{m} + \frac{m}{n}

So it would be a good idea to graph the function f(x)=4x+\frac{1}{x}.

 

[Felafel]

enlightning, thank you!

 

[HallsofIvy]

In particular, suppose n= 1 and m= 100000. What member of the set would that give?