  As mathematical induction is applied to prove statements on natural numbers,
{\it continuous induction} (or, {\it real induction}) is a tool to prove some
statements in real analysis.(Although, this comparison is somehow an
overstatement.) Here, we first consider it on densely ordered abelian groups to
prove Heine-Borel theorem (every closed and bounded interval is compact with
respect to order topology) in those structures. Then, using the recently
introduced notion of pseudo finite sets, we introduce a first order definable
version of {\it continuous induction} in the language of ordered groups and we
use it to prove a definable version of Heine-Borel theorem on densely ordered
abelian groups.
