Definable Continuous Induction on Ordered Abelian Groups

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.