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 areThus, 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 , , , and so on.

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

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

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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
.

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

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.

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