Set is Coarser than Image of Mapping of Infima

Theorem

Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.

Let $f, g:\N \to S$ be mappings such that:

$\forall n \in \N: \map g n = \inf \set {\map f m: m \in \N \land m \le n}$

Then $f \sqbrk \N$ is coarser than $g \sqbrk \N$

where $f \sqbrk \N$ denotes the image of mapping $f$.

Let $x \in f \sqbrk \N$.

By definition of image of mapping:

$\exists n \in \N: x = \map f n$

By definition of $g$:

$\map g n = \inf \set {\map f m: m \in \N \land m \le n}$

By definition of reflexivity:

$n \le n$

Then

$\map f n \in \set {\map f m: m \in \N \land m \le n}$

By definitions of infimum and lower bound:

$\map g n \preceq x$

By definition of image of mapping:

$\map g n \in g \sqbrk \N$

Thus

$\exists y \in g \sqbrk \N: y \preceq x$

$\blacksquare$