A sequence $(a_n)$ tends to $+\infty$ iff given any number $C$ there's a number $N$ such that $n > N \implies a_n \ge C.$
I'd be interested to see how this definition can be applied to the two sequences below
$(a) \; 1, 1, 2, 1, 3, 1, \ldots, n, 1, \ldots$
$(b) \; 2, 1, 4, 3, \ldots, 2n, 2n - 1,\ldots$
Thinking out loud: $n$ stands for index of a term. For $(b)$, let $C = 100000000000000000000000000$, but there's still some index $N$ among $n$ such that beyond the term indexed by $N$(the term $C$ occurs at this index or before it), all terms are greater than $C$. It's not true for $(a)$ because for any $C$ at index $N$, there's a term at index $N + k$ that equals $1$.
Do we think like that or how can I correct my thinking?