Question about sequences.

Question

A sequence can be defined as a function with the natural numbers as its domain. For instance, if $f(n)$ is a sequence then it is true that: $$f: \Bbb N \rightarrow \Bbb R$$ Therefore, from definition a sequence $f$ is an infinite "set" of real numbers in a specific order. How come then there are people talking about "finite" sequences (For example, Sal in khan academy said the term in his first Algebra I sequence video). Doesn't that contradict the definition above?

Answer 1

Good question.

You will often encounter essentially unimportant ambiguities of this kind in mathematics. The common word "sequence" sometimes means only infinite sequences, sometimes only finite, sometimes either. You can usually tell from the context. If there's any doubt, the writer should make his or her meaning clear.

Answer 2

A finite sequence (by the way, I do not use that expression) of real numbers is a function from $\{1,2\ldots,n\}$ into $\mathbb R$, for some natural number $n$.