Let $\Theta$ and $o$ be defined as usual (Landau-notation). For two equivalence classes defined by $\Theta$ we define $$\Theta(f) <_o \Theta(g) :\Leftrightarrow f \in o(g)\qquad.$$ Let $$\mathbb{F}:= \{\Theta(f)\mid f:\mathbb{N}\rightarrow\mathbb{N}\}$$
My question: Is $\langle \mathbb{F},<_o\rangle$ Dedekind-complete, i.e. given a set $F\subseteq \mathbb{F}$ with an upper bound $f_u,\forall f \in Ff\in o(f_u)$ are there $f_\inf$ and $f_\sup$ s.t. $$\forall \Theta(f)\in F: \Theta(f_\inf) <_o \Theta(f) \qquad{(1)}$$ $$\forall g: (g\text{ fulfills (1)} \Rightarrow g \in o(f_\inf))$$
($f_\sup$ is defined analogously)?
Note: We don't need to assume a lower bound, since there is no infinite strictly decreasing series of $n \in \mathbb{N}$, i.e. $\Theta(0)$ is always a lower bound.
Background: I'd like to consider $\inf \{f \mid L \in N/DTIME/SPACE(f)\}$ for some Language $L$, but that only makes sense if such a (class of) function(s) exist(s).