Set is Coarser than Image of Mapping of Infima

Theorem
Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice.

Let $f, g:\N \to S$ be mappings such that
 * $\forall n \in \N: g\left({n}\right) = \inf \left\{ {f\left({m}\right): m \in \N \land m \le n}\right\}$

Then $f\left[{\N}\right]$ is coarser than $g\left[{\N}\right]$

where $f\left[{\N}\right]$ denotes the image of mapping $f$.

Proof
Let $x \in f\left[{\N}\right]$.

By definition of image of mapping:
 * $\exists n \in \N: x = f\left({n}\right)$

By definition of $g$:
 * $g\left({n}\right) = \inf \left\{ {f\left({m}\right): m \in \N \land m \le n}\right\}$

By definition of reflexivity:
 * $n \le n$

Then
 * $f\left({n}\right) \in \left\{ {f\left({m}\right): m \in \N \land m \le n}\right\}$

By definitions of infimum and lower bound:
 * $g\left({n}\right) \preceq x$

By definition of image of mapping:
 * $g\left({n}\right) \in g\left[{\N}\right]$

Thus
 * $\exists y \in g\left[{\N}\right]: y \preceq x$