How to understand a sequence is a function?

Question

Sequence is defined as a function whose domain is N in Spivak's Calculus. But let {$a_1$, $a_2$, $a_3$, ...} be a sequence. It feels that the sequence is more like a sequence of functions rather than a single function, since $a_1$ is f(1)=$a_1$, $a_2$ is f(2)=$a_2$,..., the sequence more like ($f_1$, $f_2$,...). Then how to understand the sequence is a single function?

Answer 1

It's as you stated. You gave $f(1)$, $f(2)$ etc. So you get $f(x)$ for any natural number $x$ by plugging it in. That gives the value of the function for any input. If you do that you have made a single function. What you wrote as $f_1$ etc were just numbers, not functions.

Answer 2

Any function is, crudely speaking, a rule where you feed something in, and get something out, subject to certain niceness conditions. In this case we have a function where if you feed in the index of a term, you get that term itself back.

Answer 3

See the sequence $\{a_1,a_2,\cdots\}$ as function $f:\mathbb{N}\rightarrow \mathbb{R}$ where $f(n)=a_n$.

So, this sequence is $\{f(1),f(2),f(3),\cdots\}$.