How does Odysseus proves his identity to Penelope by .?

Odysseus proves his identity to Penelope by revealing the secret of their bed, which he constructed himself from a living olive tree, making it impossible to move. When Penelope mentions moving the bed to prepare for his arrival, he responds with indignation, demonstrating his intimate knowledge of their life together. This unique detail reassures Penelope that the man before her is indeed her husband, Odysseus, and not an imposter.