  Sequences diverge either because they head off to infinity or because they
oscillate. Part 1 constructs a non-Archimedean framework of infinite numbers
that is large enough to contain asymptotic limit points for non-oscillating
sequences that head off to infinity. It begins by defining Archimedean classes
of infinite numbers. Each class is denoted by a prototype sequence. These
prototypes are used as asymptotes for determining leading term limits of
sequences. By subtracting off leading term limits and repeating, limits are
obtained for a subset of sequences called here ``smooth sequences".
$\mathbb{I}_n$ is defined as the set of ratios of limits of smooth sequences.
It is shown that $\mathbb{I}_n$ is an ordered field that includes real,
infinite and infinitesimal numbers.
