What is the definition of supremum and how does it relate to set A?

[lukaszh]

Hello,
I found the definition. If S is supremum of set A, then
a) \forall x\in A:x\leq S
b) \forall\varepsilon>0\;\exists x_0\in A:S-\varepsilon<x_0

Now let define set A=\{1,2,3,4,5\}. Is number 5 supremum of set A? Condition a) is satisfied, but b) is problem. If \varepsilon=0.1, there isn't x_0 in the set A such that
5-0.1<x_0

Could you explain that?

 

[quasar987]

What's wrong with x_0=5 ?

 

[Tac-Tics]

As quasar said above, x0 is chosen to be 5.

The supremum of a finite set is always the same as the maximum of that set. The supremum, in a sense, can be thought of as a generalization of the idea of maximum.

 

[JG89]

The supremum of a set is either an element of the set or a limit that the elements tend to. Of course in this finite set the supremum is going to be an element of the set.

 

[HallsofIvy]

Hello,
I found the definition. If S is supremum of set A, then
a) \forall x\in A:x\leq S
b) \forall\varepsilon>0\;\exists x_0\in A:S-\varepsilon<x_0

Now let define set A=\{1,2,3,4,5\}. Is number 5 supremum of set A? Condition a) is satisfied, but b) is problem. If \varepsilon=0.1, there isn't x_0 in the set A such that
5-0.1<x_0

Could you explain that?

The explanation is that the definition you give is wrong. (a) is the definition of "upper bound". In order that an upper bound be a "supremum" (also called "least upper bound") it must be the smallest of all upper bounds. In this case, every number less than or equal to 5 is an upper bound so 5 is the "least upper bound" or "supremum".

 

[Office_Shredder]

Halls, the definition consists of both parts (a) and (b). (a) says that it's an upper bound. (b) says that it's the least such upper bound (notice that (b) doesn't make S an upper bound by itself)

 

[HallsofIvy]

But the whole point is that, while if (a) and (b) are both true, then S is a supremum, it can happen, as in the example given here, that a number is a supremum without (b) being true.