Analysis - Upper and Lower Bounds

[teme92]

Homework Statement

Let E be a non-empty subset of the real numbers R. Define carefully each of the terms
(i) 'y is an upper bound for E' and (ii) 'y is the least upper bound for E'

Homework Equations

The Attempt at a Solution

I've just started doing analysis and I'm unsure how I'm supposed to answer this. As its a definition I can't come up with any solution. Any help would be much appreciated. Thanks everyone

 

[Ray Vickson]

Homework Statement

Let E be a non-empty subset of the real numbers R. Define carefully each of the terms
(i) 'y is an upper bound for E' and (ii) 'y is the least upper bound for E'

Homework Equations

The Attempt at a Solution

I've just started doing analysis and I'm unsure how I'm supposed to answer this. As its a definition I can't come up with any solution. Any help would be much appreciated. Thanks everyone

You will have go struggle with it---we are not allowed to answer for you. Are you using a textbook? Do you have course notes? Are you claiming that those sources contain nothing on the relevant topics?

 

[teme92]

Hi Ray,

I understand what upper and lower bounds are its just the phrasing of the question that's confusing me. Can I say:

E={x ε R: x≥n & x≤m, n,m εR}

Y is upper bound if y≥n
Y is lower bound if y≤m

Thanks for the quick response and appreciate any further help.

 

[Ray Vickson]

Hi Ray,

I understand what upper and lower bounds are its just the phrasing of the question that's confusing me. Can I say:

E={x ε R: x≥n & x≤m, n,m εR}

Y is upper bound if y≥n
Y is lower bound if y≤m

Thanks for the quick response and appreciate any further help.

No, you have it backwards. The way you have written E, it must be the interval [n,m], but that was not given in the question. All you know is that $E \subset \mathbb{R}$. For example, E could be all the real numbers of the form $1-1/n, n = 1,2, \ldots$, and these certainly do not form an interval.

In the lines below, where you say "Y is upper bound if y≥n" etc, you do not say what is n.

 

[teme92]

So do I just say:

i) yεF is an upper bound for E if x≤y whenever xεE

ii) yεF is a least upper bound if y is an upper bound for E and if y1εF and y1≤y then y1 is not an upper bound for E