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 $f$.

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