I have read a proof of the titled theorem, where P(n) is the number of partitions of the natural number n. The author I have read is G.E. Andrews in his fine but (for me) challenging monograph (Dover, 1971). Prof. Andrews's proof is on p. 213 (Appendix A).
Rather than ask for someone to detail the above referenced proof for me more expansively, I would like to be referred to another proof of the theorem, a proof that I might find more readable.
By the way, my title should have stated in the limit as n gets very large.