I have the following definition in my book, and was confused as to the context of the word "rank" here. The definition is as follows:
A sequence $(u_n)_{n∈N}$ has limit $l ∈ R$ as $n → ∞$ (we also say “converges to $l$ as $n → ∞$”) if, for any $ε > 0$, there exists a rank $N_ε ∈ N$ such that, for any $n ≥ N_ε$, $|u_n − l| ≤ ε$...
I understand everything except for the portion "there exists a rank $N_ε ∈ N$".
What exactly does this line mean to convey?