Which is the best notation for a sequence?

Question

In a set, the order of its elements is (as far as I know) not important; in a sequence, the order of its elements is important.

Which is the notation I should use in order to define a sequence? I could denote the sequence of positive integers by $S$.

Now, suppose I want to list (in order) the terms of sequence $S$. I'm not sure if the notation

$$S=\{1, 2, 3, 4, ...\}$$

is appropriate. Besides, I'm not sure if using $=$ is appropriate.

Any comments?

Answer 1

You can denote the sequence by $$S = \{a_k\}_{k = 1}^\infty $$ where $a_k = k$.