Infimum and supremum

Printable View

February 8th 2010, 04:21 PM
summerset353

Infimum and supremum

I dont understand the infimum and supremum. Could someone please explain these terms. I have the definitions and don't quite understand the full meaning. Maybe explain using an example such as proving the infimum and supremum on the interval (a,b). An example always helps.
February 8th 2010, 04:40 PM
Nyrox

I'll only try to give intuition for the supremum, and in $\mathbb R$ . The infimum is pretty much the same deal :p

Imagine you have a set $X$ in $\mathbb R$ . It is legitimate to ask whether this set has a maximum element or not, that is: is there an element $s$ , belonging to $X$ , such that $s\geq x$ for all other $x\in X$ ? This element may or may not exist.

Example 1: $[a,b]$ , where $a<b$ has a maximum element: $b$ .

Example 2: $]a,b[$ , does not have a maximum element. You can get as close to $b$ as you wish, but there is no "end point".

Example 3: $]a,\infty[$ does not have a maximum element.

Now let's take a closer look at example 2. If we were to define a "generalized maximum" concept, it would be extremely tempting to say the "generalized maximum" of $]a,b[$ is $b$ , since, as we've stated, we can get as close to $b$ as we wish... we're just... missing the point itself!

What property then, does $b$ have, that makes us want to choose it as the "generalized maximum"? And, since we want to generalize the notion of maximum, we want to define this new concept in such a way that $b$ is also a "generalized maximum" in example 1.

Instead of thinking of the intervals $[a,b]$ and $]a,b[$ , we then think of the set of all upper bounds of $[a,b]$ and $]a,b$ respectively, and observe that, in both cases, $b$ is the minimum of the corresponding sets of upper bounds. We take that as a definition: we say that an element $b$ is a supremum of a set $X$ if $b$ is the least upper bound of $X$ .

And of course, we may think of the supremum of $X$ as a value that is bigger than all elements of $X$ , and is "infinitely" close to $X$ . It may, or may not belong to $X$ itself. If it does belong, we are in the case of example 1: the set has a maximum. If it doesn't, we're in the case of example 2.

Finally observe that example 3 does not have a supremum either. There is no fixed value that is greater than all others of the interval.

Let me know if it was of any help ^^
February 8th 2010, 04:57 PM
mabruka

Very good answer indeed.
February 8th 2010, 05:42 PM
summerset353

Thank you that really helps!
February 9th 2010, 04:57 PM
xalk

Quote:

Originally Posted by summerset353

I dont understand the infimum and supremum. Could someone please explain these terms. I have the definitions and don't quite understand the full meaning. Maybe explain using an example such as proving the infimum and supremum on the interval (a,b). An example always helps.

The official definition for the supremum is the following:

If S is a set such that $S\neq\emptyset$ and $S\ subset R$ then,

.................SupS = v iff..............

1) for all,x: if xεS ,then $x\leq v$

2) for all ε>0 there exists a yεS ,such that v-ε<y $\leq v$

The first condition tell us that v is an upper bound and the second that v is the least upper bound.

Now can you use the above fefinition and prove that Sup(a,b) = b??

The definition for infemum is similar