$\mathbb{R}$ as set of Dedekind cuts on $\mathbb{Q}$

$\mathbb{R}$ as set of Dedekind cuts on $\mathbb{Q}$

-- "let $A,B \in \mathbb{Q}$, $A < B$ if $A\leq B$ and $A \neq B$
-- "let $\preceq$ be a ordering of a set $A$, and $B \subsetneqq A$, $B$
is initial segment of $A$ under $\preceq $ if $\forall a \in A, \forall b
\in B(a \preceq b \to a \in B )$"
the following definitions are correct:
-- "let $\leq$ be a ordering of $\mathbb{Q}$, and $B \subsetneqq
\mathbb{Q}$, $B$ is Dedekind cut on $\mathbb{Q}$ if $B$ initial segment of
$A$ under $\leq$ and $\forall C \in B, \exists D \in B (C < D)$"
-- "$\mathbb{R}:=\{B \subsetneqq \mathbb{Q}|B\text{ is Dedekind cuts on }
\mathbb{Q}\}$"
??
Thanks in advance!!