The nurse tells Penelope that it really is Odysseus. Penelope doesn't believe it and thinks it's a trick of the gods. The nurse tries to give her proof it's him and says to her
"I can give you another proof; when I was washing him I perceived the scar which the wild boar gave him, and I wanted to tell you about it, but in his wisdom he would not let me, and clapped his hands over my mouth; so come with me and I will make this bargain with you- if I am deceiving you, you may have me killed by the most cruel death you can think of."
Then Telemachus chastises her for not believing him, either. He sees that she is cold to him and doesn't even try to accept him back after all of those years. He says that most women would sit next to him and ask questions. He says that her heart must be of stone. These comments/proofs should make her believe--it's only until she gets the information from him that proves to HER that it really is Odysseus that she lets down her guard.