 Sorted multiplicative closure stream

For a finite nonempty collection of exact real multipliers greater than one, the multiplicative closure of one can be enumerated by a shared lazy list: emit one, then merge its elementwise multiples by all generators. Every value after one has a smaller predecessor obtained by removing one factor, proving productivity by increasing value. There are only finitely many exponent tuples below any fixed bound, because every generator is greater than one; thus locally finite sorted-stream merge eventually emits every generated value. Equal products are emitted once.

A multiplier below one contradicts increasing order starting at one. A multiplier equal to one adds no new values and should be removed from the recursive streams; an empty remaining generator collection has the finite closure consisting of one. These qualifications concern exact arithmetic: finite precision or overflow can violate the order and growth assumptions.