The infimum and supremum are concepts in mathematical analysis that generalize the notions of minimum and maximum of finite sets. They are extensively used in real analysis, including the axiomatic construction of the real numbers and the formal definition of the Riemann integral. The limits of the infimum and supremum of parts of sequences of real numbers are used in some convergence tests and, in particular, in computations of domains of convergence of power series.
Let be a subset of the real numbers . Then is the greatest lower bound of the elements of , if it exists; that is, is the largest real number such that for all . Similarly, is the least upper bound of the elements of , if it exists; that is, is the smallest real number such that for all .
(1) If is finite, then and are just the minimum and maximum elements of respectively.
(2) Note that and may not lie in in general. For instance, if , then and
(3) The infimum and supremum are unique if they exist.
(4) If does not have a lower bound, it is reasonable to write If does not have an upper bound, it is reasonable to write
(5) The infimum and supremum are related via where This is often convenient for proofs.
The following property is a useful characterization of the infimum and supremum of a set of real numbers.
Let be a set of real numbers.
Suppose is a lower bound for Then if and only if, for every , there is an such that .
Suppose is an upper bound for Then if and only if, for every , there is an such that .
The proofs of the two statements are more or less identical and can be formally translated to each other by remark (5) above Here is the proof of the first statement. If then cannot be a lower bound for , so there must be an element of that is bigger than it. On the other hand, if is a lower bound that is not the infimum, then there is a larger lower bound for . Let ; then there is no such that
The concepts of infimum and supremum can be extended to functions on the real numbers:
If is a real-valued function defined on a subset of the real numbers, define
(The proof is left as an exercise.)
Constructing the real numbers from scratch is a standard topic in introductory analysis. One way is to start with the integers, then create the rational numbers, and then pass to the real numbers by viewing them as limits of certain types of sequences of rational numbers. This process is called completion, as in " is the completion of ."
The fundamental property that the real numbers satisfy is called completeness. There are several formulations of this property that are logically equivalent. One of them is the least upper bound property:
Every nonempty subset of the real numbers with an upper bound has a supremum.
If one uses the notation for sets with no upper bound as in remark (4) above, this can be restated "every nonempty subset of the real numbers has a supremum (which may be )."
Note that this property is not true for the rational numbers: the set of all rational numbers less than has an upper bound that is rational (e.g. 2), but there is no least rational upper bound there are rational numbers less than for any
The least upper bound property implies many of the basic facts about the real numbers that are used in analysis.
The intermediate value theorem states that if is a continuous function on and is any number between and then there is some such that
To see that this theorem follows from the least upper bound property, suppose without loss of generality that and consider Then is nonempty since and it has an upper bound namely so there is a least upper bound. Call that least upper bound
Suppose Then let By continuity, there is a such that implies But implies for all in that range, so no 's in that range lie in So is an upper bound for as well, which is a contradiction of the "leastness" of
Suppose . Then let By continuity, there is a such that implies But implies for all in that range, so every in that range lies in So, for instance, is in , which is a contradiction since is an upper bound.
The conclusion is that as desired.
Let be a sequence of real numbers. For any let Then
Let be the sequence . Then where and where The limit of both of these expressions as is which is also the limit of the sequence. So
Note that and
(1) Unlike the limit of the sequence, the and always exist, if we allow and as possible values. This is because the sequence is a non-decreasing sequence similarly the sequence is non-increasing so its limit either exists or equals .
(2) The limit exists if and only if if the limit exists, all three values are equal.
(3) If then it is the smallest real number such that, for any only finitely many elements of the sequence are Note that it may not be the case that only finitely many elements of the sequence are For instance, So "every number larger than the is an eventual upper bound." Similarly, every number smaller than the is an eventual lower bound.
Let be the prime number. Then
Proof: are all composite, for any So for any there are infinitely many values of such that . Take to be the largest prime less than , where ; then the next prime is at least integers away. So cannot be an eventual upper bound, so it is not larger than the Since this is true for all , the result follows.
On the other hand,
is still unknown. The twin primes conjecture is equivalent to the statement that it equals but currently all that is known is that it is at most (Until 2013, it was not even known that it was finite!)
Let be a power series with complex coefficients. Let
If the denominator is let If the denominator is let Then the series converges if and diverges if . If the series always converges.
In this context, is called the "radius of convergence" for the series.
has radius of convergence because equals