Paracompactness and the Lindelöf property in countable products

In this paper, we prove the following: Let Y be a perfect paracompact (hereditarily Lindelöf) space and {Xn:n∈ω} be a countable collection of Čech-scattered paracompact (Lindelöf) spaces, then the product Y×∏n∈ωXn is paracompact (Lindelöf).