Definition of initial segment

Definition of initial segment

the definition of initial segment is correct:
"let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$,
$B$ is initial segment of $A$ under $\preceq$ if $\forall a \in A, \forall
b \in B(a \preceq b \to a \in B )$"
???
Thanks in advance!!