Mathematical induction in form of - "in well-founded poset any progressive subset is total" (by progressive of $S$ i mean that $\forall x \ ((\forall a<x \ (a \in S)) \rightarrow x \in S)$) somhow looks to me similar to completness of metric space in form of - "cover of closed bounded subspace has finite subcover". Well-foundedness looks to me similar to boundedness and closeness, and totality of progressive set looks similar to existance of finite subcover. Sometimes proof by supremum in real numbers looks very similar to proof by induction in natural numbers. I don't know how to strictly formulate it, and can't find anything about it.
Is there some generalization of those principles, or another link between them?
