Definition:Sequence/Infinite Sequence

Definition

An infinite sequence is a sequence whose domain is infinite.

That is, an infinite sequence is a sequence that has infinitely many terms.

Hence for an infinite sequence $\sequence {s_n}_{n \mathop \in \N}$ whose range is $S$, $\sequence {s_n}_{n \mathop \in \N}$ is an element of the set of mappings $S^{\N}$ from $\N$ to $S$.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ all sequences are infinite sequence unless explicitly specified otherwise.

Also known as

Treatments of infinite sequences which approach the definition of a sequence from the direction of ordered tuples offer the term $\omega$-tuple for the concept of a tuple whose domain is (countably) infinite.

Some sources use the term denumerable sequence.

Also see

• Results about infinite sequences can be found here.

Sources

• 1958: J.A. Green: Sequences and Series ... (previous) ... (next): Chapter $1$: Sequences: $1$. Infinite Sequences
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.15$: Sequences: Definition $15.1$
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 10$: Arbitrary Products: Exercise $4$
• 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2$: Infinite Series of Constants
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.12$: Induction
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): infinite sequence
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): sequence
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): infinite sequence
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): sequence
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 8$ Definition by finite recursion
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.3$: Definition $\text{A}.12$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): infinite sequence