What leads algernon to suspect that ernest/jack is a bunburyist?
*algernon finds a handkerchief embroidered "jack worthing" in his friends jacket pocket.
*gwendolen tells him that his name is really jack.
*he overhears him call himself jack to a friend.
*he sees an inscription on a cigarette case.

Well the story says that he sees an inscription on a cigarrete case.We need to have in mind that  Algernon also created a "double" whose name is "Bunbury". Bunbury lives in the country. When Algernon wants to get out of things he does not feel like doing, he goes to visit his fake friend "Bunbury" in the country as an excuse. Algernon thinks that "ERNEST" provides the same "double" for Jack.