How can we have more than one sequence in a set? Code Answer

I am looking at a proof for a divergence criterion for functional limits:

Let $f$ be a function defined on $A$, and let $c$ be a limit point of $A$. If there exist two sequences $(x_n)$ and $(y_n)$ in $A$ with $x_n neq c$ and $y_n neq c$ and $lim x_n= lim y_n=c$ but $lim f(x_n)= lim f(y_n)$,then we can conclude that the functional limit $lim_{x→ c}f(c)$does not exist.

Okay, now that that is out of the way, and I might just be getting terminilogy mixed up, but
say we take the set $A$ to be $[1,10] in mathbb{N} $

Thus isn’t the only sequence $1,…,10$ ? and any other combination just a subsequence?

Is a Sub-sequence considered a “sequence” in this definition for divergence?

Answer

A sequence:

1. is ordered, meaning that it matters in which order the elements occur
2. can contain elements multiple times

So a sequence in the set $A$ might be

$$(1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,….)$$

Or

$$(1,2,1,2,1,2,1,2,….)$$

Or

$$(1,1,1,1,1,1,…..)$$

But also just

$$(1)$$

Or

$$(1,5,10)$$

And yes a subsequence is always by definition also a sequence.