Understanding Inf & Sup in Riemann Sums

[Bashyboy]

I am currently reading about riemann sums and several different sources uses these abbreviations, inf and sup, and I am not certain what they mean. Could someone explain them to me?

 

[micromass]

They are the infimum and the supremum.

The supremum of a set A is the least upper bound and the infimum is the greatest lower bound.

For example, consider the set [0,1]. This has many upper bounds. For example, 2, 10, 10020330 are all upper bounds. But 1 is the smallest such upper bound. Thus 1 is the supremum.

In the previous example, 1 was actually a maximum: that is, the greatest element contained in the set. But a supremum does not need to be contained in the set. For example, ]0,1[ (or (0,1) in other notation) also has 1 as smallest upper bound. Every element smaller than 1 will not be an upper bound anymore. Thus 1 is the supremum of the set.

The same discussion holds for infima.
For example: $$inf ]0,2[ = 0$$ or somewhat more complicated $$inf \{1/n~\vert~n\in \mathbb{N}\}=0$$ (note that I take $0\in \mathbb{N}$).

 

[Bashyboy]

So, are the words, in a way, synonymous to minimum and maximum?

 

[Bashyboy]

Also, since this particular thread pertains to riemann sums, is the reason why the definite integral is defined as the limit of the riemann sum simply because they produce the same result, or is there some deeper meaning? And what was the definite integral defined as before Bernhard Riemann came along?

 

[micromass]

So, are the words, in a way, synonymous to minimum and maximum?

No! That was the entire point. The supremum is a generalization of the maximum.
When I say that an element x is the maximum of A, then this means that x is the greatest element contained in A. So it is implied that x is an element of A.
But with the supremum, x does not need to be an element of A. For example sup ]0,1[ = 1, but 1 is not an element of ]0,1[.

 

[HallsofIvy]

For example, the open interval, (0, 1)= {x| 0< x< 1}, does not have a "minimum" or maximum. But its inf (also called "greatest lower bound") is 0 and its sup (also called "least upper bound") is 1.

If a set has a "minimum" then its infimum is that minimum. Similarly, if a a set has a "maximum" then its supremum is that maximum. But any set with a lower bound has an infimum but not necessarily a "minimum" and any set with an upper bound has a supremum but not necessarily a "maximum".

(Some texts allow "$-\infty$" as an infimum and "$\infty$" as a supremum so that a set does not have to be bounded to have infimum and supremum.)