the definition of Dedekind cut by initial segment is correct:
"let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B \neq \emptyset$, $B$ is Dedekind cut of $A$ under $\preceq$ if $B$ is initial segment of $A$ under $\preceq$ and $B$ has no largest element under $\preceq$"
???
Thanks in advance!
P.S.= in this case "$B$ has no largest element under $\preceq$" means "$\forall a \in B, \exists b \in B(b \preceq a \wedge b\neq a)$"
If you're thinking of the construction of the reals from the rationals, you need to add two more conditions: $B\ne\varnothing$, and $B\ne A$. If you don't add these conditions, you get something order-isomorphic to $[0,1]$, i.e., to the extended reals (with $\pm\infty$). If you're thinking of the construction of the Dedekind completion of a linear order, what you have is fine.