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From the entire set of natural numbers successively deleting some residue classes mod a prime, we invent a recursive sieve method. This is a modulo algorithm on natural numbers and their sets. The recursively sifting process mechanically yields a sequence of sets, which converges to the set of the certain subscripts of Fibonacci composites. The corresponding cardinal sequence is strictly increasing. Then the well known theory, set valued analysis, allows us to prove that the set of the certain subscripts is an infinite set, namely, the set of Fibonacci composites with prime subscripts is infinite. 

Fibonacci composites, recursive sieve method, order topology, limit of sequence of sets.