Sequences

Let be a set of objects (e.g., real numbers, events, random variables). A sequence of elements of is a function from the set of natural numbers to the set , i.e., a correspondence that associates one and only one element of to each natural number . In other words, a sequence of elements of is an ordered list of elements of , where the ordering is provided by the natural numbers.

A sequence is usually indicated by enclosing a generic element of the sequence in curly brackets:where is the -th element of the sequence. Alternative notations are [eq2] Thus, if is a sequence, is its first element, is its second element, is its -th element, and so on.

Example Define a sequence by characterizing its -th element as follows: is a sequence of rational numbers. The elements of the sequence are $a_{1}=1$ , $frac{1}{2}$ , $frac{1}{3}$ , $frac{1}{4}$ and so on.

Example Define a sequence by characterizing its -th element as follows: [eq8] is a sequence of and . The elements of the sequence are $a_{1}=0$ , $a_{2}=1$ , $a_{3}=0$ , $a_{4}=1$ and so on.

Example Define a sequence by characterizing its -th element as follows: [eq11] is a sequence of closed subintervals of the interval . The elements of the sequence are , , , and so on.

Table of contents

Countable and uncountable sets
Limit of a sequence

Countable and uncountable sets

Let be a set of objects. is a countable set if all its elements can be arranged into a sequence, i.e., if there exists a sequence such thatIn other words, is a countable set if there exists at least one sequence such that every element of belongs to the sequence. is an uncountable set if such a sequence does not exist. The most important example of an uncountable set is the set of real numbers .

Limit of a sequence

The concept of limit of a sequence is discussed in the lecture entitled Limit of a sequence.

How to cite

Please cite as:

Taboga, Marco (2021). "Sequences", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/mathematical-tools/sequences.