In this lecture we introduce the notion of limit of a sequence . We start from the simple case in which is a sequence of real numbers, then we deal with the general case in which can be a sequence of objects that are not necessarily real numbers.
We first give an informal definition and then a more formal definition of the limit of a sequence of real numbers.
Let be a sequence of real numbers. Let . Denote by a subsequence of obtained by dropping the first terms of , i.e.,The following is an intuitive definition of limit of a sequence.
Definition (informal) Let be a real number. We say that is a limit of a sequence of real numbers if, by appropriately choosing , the distance between and any term of the subsequence can be made as close to zero as we like. If is a limit of the sequence , we say that the sequence is a convergent sequence and that it converges to . We indicate the fact that is a limit of by
Thus, is a limit of if, by dropping a sufficiently high number of initial terms of , we can make the remaining terms of as close to as we like. Intuitively, is a limit of if becomes closer and closer to by letting go to infinity.
The distance between two real numbers is the absolute value of their difference. For example, if and is a term of a sequence , the distance between and , denoted by , isBy using the concept of distance, the above informal definition can be made rigorous.
Definition (formal) Let . We say that is a limit of a sequence of real numbers ifIf is a limit of the sequence , we say that the sequence is a convergent sequence and that it converges to . We indicate the fact that is a limit of by
For those unfamiliar with the universal quantifiers (any) and (exists), the notation reads as follows: "For any arbitrarily small number , there exists a natural number such that the distance between and is less than for all the terms with ", which can also be restated as "For any arbitrarily small number , you can find a subsequence such that the distance between any term of the subsequence and is less than " or as "By dropping a sufficiently high number of initial terms of , you can make the remaining terms as close to as you wish".
It is also possible to prove that a convergent sequence has a unique limit, i.e., if has a limit , then is the unique limit of .
Example Define a sequence by characterizing its -th element as follows:The elements of the sequence are , , , and so on. The higher is, the smaller is and the closer it gets to . Therefore, intuitively, the limit of the sequence should be :It is straightforward to prove that is indeed a limit of by using the above definition. Choose any . We need to find an such that all terms of the subsequence have distance from zero less than :Note first that the distance between a generic term of the sequence and iswhere the last equality obtains from the fact that all the terms of the sequence are positive (hence they are equal to their absolute values). Therefore, we need to find an such that all terms of the subsequence satisfySincethe conditionis satisfied if , which is equivalent to . Therefore, it suffices to pick any such that to satisfy the conditionIn summary, we have just shown that, for any , we are able to find such that all terms of the subsequence have distance from zero less than . As a consequence is the limit of the sequence .
We now deal with the more general case in which the terms of the sequence are not necessarily real numbers. As before, we first give an informal definition, then a more formal one.
Let be a set of objects (e.g., real numbers, events, random variables) and let be a sequence of elements of . The limit of is defined as follows.
Definition (informal) Let . We say that is a limit of a sequence of elements of , if, by appropriately choosing , the distance between and any term of the subsequence can be made as close to zero as we like. If is a limit of the sequence , we say that the sequence is a convergent sequence and that it converges to . We indicate the fact that is a limit of by
The definition is the same we gave above, except for the fact that now both and the terms of the sequence belong to a generic set of objects .
In the definition above, we have implicitly assumed that the concept of distance between elements of is well-defined. Thus, for the above definition to make any sense, we need to properly define distance.
We need a function that associates to any couple of elements of a real number measuring how far these two elements are. For example, if and are two elements of , needs to be a real number measuring the distance between and .
A function is considered a valid distance function (and it is called a metric on ) if it satisfies some properties, listed in the next proposition.
Definition Let be a set of objects. Let . is considered a valid distance function (in which case it is called a metric on ) if, for any , and belonging to :
non-negativity: ;
identity of indiscernibles: if and only if ;
symmetry: ;
triangle inequality: .
All four properties are very intuitive: property 1) says that the distance between two points cannot be a negative number; property 2) says that the distance between two points is zero if and only if the two points coincide; property 3) says that the distance from to is the same as the distance from to ; property 4) says (roughly speaking) that the distance you cover when you go from to directly is less than (or equal to) the distance you cover when you go from to passing from a third point (if is not on the way from to you are increasing the distance covered).
Example (Euclidean distance) Consider the set of -dimensional real vectors . The metric usually employed to measure the distance between elements of is the so-called Euclidean distance. If and are two vectors belonging to , then their Euclidean distance iswhere are the components of and are the components of . It is possible to prove that the Euclidean distance satisfies all the four properties that a metric needs to satisfy. Furthermore, when , it becomeswhich coincides with the definition of distance between real numbers already given above.
Whenever we are faced with a sequence of objects and we want to assess whether it is convergent, we need to first define a distance function on the set of objects to which the terms of the sequence belong and verify that the proposed distance function satisfies all the properties of a proper distance function (a metric). For example, in probability theory and statistics, we often deal with sequences of random variables. To assess whether these sequences are convergent, we need to define a metric to measure the distance between two random variables. As we will see in the lecture entitled Sequences of random variables and their convergence, there are several ways of defining the concept of distance between two random variables. All these ways are legitimate and are useful in different situations.
Having defined the concept of a metric, we are now ready to state the formal definition of a limit of a sequence.
Definition (formal) Let be a set of objects. Let be a metric on . We say that is a limit of a sequence of objects belonging to ifIf is a limit of the sequence , we say that the sequence is a convergent sequence and that it converges to . We indicate the fact that is a limit of by
Also in this case, it is possible to prove (see below) that a convergent sequence has a unique limit, i.e., if has a limit , then is the unique limit of .
The proof is by contradiction. Suppose that and are two limits of a sequence and . By combining property 1) and 2) of a metric (see above) it must be thati.e., where is a strictly positive constant. Pick any term of the sequence. By property 4) of a metric (the triangle inequality), we haveConsidering that , the previous inequality becomesNow, take any . Since is a limit of the sequence, we can find such that , which means thatandTherefore, can not be made smaller than and as a consequence cannot be a limit of the sequence.
In practice, it is usually difficult to assess the convergence of a sequence using the above definition. Instead, convergence can be assessed using the following criterion.
Lemma (criterion for convergence) Let be a set of objects. Let be a metric on . Let be a sequence of objects belonging to and . converges to if and only if
This is easily proved by defining a sequence of real numbers whose generic term isand noting that the definition of convergence of to , which iscan be written aswhich is the definition of convergence of to .
So, in practice, the problem of assessing the convergence of a generic sequence of objects is simplified as follows:
find a metric to measure the distance between the terms of the sequence and the candidate limit ;
define a new sequence , where ;
study the convergence of the sequence , which is a simple problem, because is a sequence of real numbers.
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