I am looking to see if anyone could help to conform to me if my intuition is correct, or if not please explain how I can understand it better. For example, if I wanted to say find the sup, inf of the following set S,
$$S=\{\frac{m}{m+n} : m , n \in \mathbb{N} \}$$
and we are to be considering $$\mathbb{N}=\{1,2,…\}$$
My first though that I am wondering is , is it correct to say
Suppose that sup S exists and say it is denoted as $supS=u$
then it must satisfy $$u-\epsilon \lt S{\epsilon}$$ for any $$\epsilon \gt 0$$, and $S_{\epsilon} \in S$ ( i.e. if we can prove this for some u, then that u is in fact the sup of S?)
and similarly, a number $\alpha$ is the infimium iff it is such that
it must be that $$\alpha+\epsilon > S_{\epsilon}$$ for any $$\epsilon \gt 0 $$
I hope so far that makes some sense.
In terms of the values,
I am not to sure, I see that if we had m=1, then n could always just get higher and we would tend to a value of 0.
if n=1, and m=1 we could get $\frac{1}{2}$ , but if we keep n=1 and allows m to get higher then we could approach 1 Could anyone help to me finish off details, if the method is true?
How could I formulate it in terms of epsilon etc.
Thank you
Your set S is not empty and has no largest or smallest member.If I understand your notation,you ask whether the property (P): Every number less than u is less than some member of S.Well,one possibility is that u itself is less than some member of S, so u is not an upper bound for S (let alone the least one). If u has property (P), then u=lub(s) only when u is the LARGEST number with property(P).Also, note that if a set S has a largest member x then x=lub(S).It is a defining property of the "reals" that if a non-empty set has an upper bound (or a lower bound) it has a least one (or a largest one).A consequence of this is that no positive real is less than every positive rational, and no real which is less than one can be greater than all of the rationals that are less than one.Hence,for your set S, no positive real is a lower bound,but all the non-negative reals are lower bounds, and the greatest of these is zero, i.e. 0=inf(S).