Proposition 3

Every monotonic sequence of real numbers has a limit in $\overline{\mathbf{R}}$. In particular, every increasing (resp. decreasing) sequence of finite numbers converges to a finite real number if it is bounded above (resp. bounded below) and to $+\infty$ (resp. $-\infty$) otherwise. For example, the sequence of positive integers converges to $+\infty$.

This fact is the origin of the notation $\lim_{n \to \infty} u_n$ to denote the limit of a sequence (Chapter I, § 7, no. 3).

Likewise, every strictly increasing sequence of integers $(p_n)$ converges to $+\infty$; for we see by induction that $p_n \geq p_0 + n$ for all $n$.