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

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?

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.