We address the problem of proving a loop invariant property within a perpetual loop. We have two goals. Our first goal is to prove the property holds at over all iterations, ie. to have unbounded verification. Failing this, our subsequent goal is to determine a loop iteration bound where the property holds, ie. to have the best possible bounded verification. Our framework is set in a harness which is essentially a one loop program whose body comprises a bounded computation, for example, one which does not contain any loops. Our interpreter is based on iterative deepening; it performs bounded reasoning at each iteration which increases the bound, and has two key features: incrementality, ie. it learns and exploits the result of the previous iteration, and induction, ie. it has a fixpoint-checking mechanism which can detect that the property is invariant throughout all iterations.