1915 both companies listed in the New York City Directory
1920 both " " at the same address in the New York City Directory
1925 both " " " " " " " " " " " "
Woodside Sterling Co is listed as late as 1965 in the New York City Directory.
I don't believe this information conflicts with "Rainwaters" except that instead of Woodside succeeding Richard M Woods c1920, the two companies coexisted at least for the period shown and possibly beyond those years as well.