infimum and supremum for real numbers
is an upper bound for , that is, for all ,
if is another upper bound for , then .
Such a number is called the supremum of , and it is denoted by . It is easy to see that there can be only one least upper bound. If and are two least upper bounds for . Then and , and .
Next, let us consider a set that is bounded from below. That is, for some we have for all . Then we say that is a a greatest lower bound for if
is an lower bound for , that is, for all ,
if is another lower bound for , then .
Such a number is called the infimum of , and it is denoted by . Just as we proved that the supremum is unique, one can also show that the infimum is unique. The next lemma shows that the infimum exists.
Every non-empty set bounded from below has a greatest lower bound.
Let be a lower bound for non-empty set . In other words, for all . Let
Let us recall the following result from this page (http://planetmath.org/InequalityForRealNumbers); if is an upper(lower) bound for , then is a lower(upper) bound for .
Thus is bounded from above by . It follows that has a least upper bound . Now is a greatest lower bound for . First, by the result, it is a lower bound for . Second, if is a lower bound for , then is a upper bound for , and , or . ∎
The proof shows that if is non-empty and bounded from below, then
In consequence, if is bounded from above, then
In many respects, the supremum and infimum are similar to the maximum and minimum, or the largest and smallest element in a set. However, it is important to notice that the and do not need to belong to . (See examples below.)
For example, consider the set of negative real numbers
Then . Indeed. First, for all , and if for all , then .
|Title||infimum and supremum for real numbers|
|Date of creation||2013-03-22 15:41:42|
|Last modified on||2013-03-22 15:41:42|
|Last modified by||matte (1858)|