applicability: $\mathbb {N,Z, Q, R}$
A divergent real series \((x_n)_{n\in\mathbb N}\) is said to tend to infinity (respectively to tend to minus infinity) if for every real constant \(C\in\mathbb R\) there exists an index \(N( C)\in\mathbb N\) with
\[x_n > C\quad\text{(respectively }x_n < C\text{)}\quad\text{for all }n\ge N( C).\]
If any of the above conditions is fulfilled, we write
\[\lim_{n\to\infty}x_n =\infty\quad\text{(respectively }\lim_{n\to\infty}x_n =-\infty\text{)}.\]
Definitions: 1 2
Lemmas: 3
Proofs: 4 5 6
Propositions: 7 8 9 10 11 12