# Thread: Help finding suprema and infima

1. ## Help finding suprema and infima

Q: Compute, without proofs, the suprema and infima of the following sets:

(a) $\{n\in\mathbb{N}:n^{2}<10\}$

(b) $\{\frac{n}{n+m}:m,n\in\mathbb{N}\}$

(c) $\{\frac{n}{2n+1}:n\in\mathbb{N}\}$

(d) $\{\frac{n}{m}:m,n\in\mathbb{N}$ with $m+n\leq\\10\}$

A:

(a) $\{n\in\mathbb{N}:n^{2}<10\}=[0,3]$. So, the Sup(A)=3 and Inf(A)=0

(b) I think the interval is $(0,1)$, but I am not sure.

(c) $\{\frac{n}{2n+1}:n\in\mathbb{N}\}=[\frac{1}{3},\frac{1}{2})$. So, Sup(A)=1/2 and Inf(A)=1/3

(d) $\{\frac{n}{m}:m,n\in\mathbb{N}$ with $m+n\leq\\10\}$. So, Sup(A)=9 and Inf(A)=1/9

I am confused by (b) mostly, but I am not sure about the rest of them either. Do the sup and inf have to be natrual numbers?

2. Originally Posted by Danneedshelp
Do the sup and inf have to be natrual numbers?
In this context I don't believe they do - but it has not been specified in what set they are allowed to be. In this context it ought to be $\mathbb Q$ (the rationals).

So I would say that for b the sup is 1 and inf is 0 as you suggest.

For the sup, set m=0 and let n be anything you like, you've got 1 and nothing you can do can make it bigger.

Second, set n = 0 and let m = anything you like, you've got 0.

Are you using the convention that $\mathbb N = \{1, 2, 3, ...\}$ or $\mathbb N = \{0, 1, 2, ...\}$? In the first case the answers will be the same but you won't have 0 and 1 actually in the sets described.

3. Originally Posted by Matt Westwood
In this context I don't believe they do - but it has not been specified in what set they are allowed to be. In this context it ought to be $\mathbb Q$ (the rationals).

So I would say that for b the sup is 1 and inf is 0 as you suggest.

For the sup, set m=0 and let n be anything you like, you've got 1 and nothing you can do can make it bigger.

Second, set n = 0 and let m = anything you like, you've got 0.

Are you using the convention that $\mathbb N = \{1, 2, 3, ...\}$ or $\mathbb N = \{0, 1, 2, ...\}$? In the first case the answers will be the same but you won't have 0 and 1 actually in the sets described.
$\mathbb N = \{1, 2, 3, ...\}$

So the first one goes from 1 to 3?

How do the others look?

Thanks a lot.

4. They look fine to me. You're all right, you know what you're doing here.