Final answer:
Hamlet offers proof of his sanity by sharing his encounter with the ghost of his father and his determination to seek revenge for his murder.
Step-by-step explanation:
In Act II, Hamlet offers proof of his sanity when he tells Horatio about the recent experience they had with the ghost of his father. He explains that they saw the ghost and it confirmed their suspicions about his uncle Claudius. Hamlet believes that the ghost is an omen of danger and deceit, and he vows to take revenge for his father's murder. This evidence demonstrates Hamlet's rational thinking and his awareness of the political situation in the kingdom.