In Act I, Scene 1 of Hamlet, how does Horatio show that he also believes in ghosts after the ghost's appearance?
We can tell that Horatio believes in ghosts—or, at least, that he has come to believe in them—by inference from what he says and does after the ghost appears in act 1, scene 1 of the play. At first, Marcellus states that Horatio "will not let belief take hold of him" and is unwilling to believe the reports he had heard from others. As such, Marcellus asks Horatio to accompany them, "that he may approve our eyes."
When the ghost appears, however, Horatio seems to change his mind. He agrees that it looks "most like" the king and states that it fills him with "fear and wonder." When the ghost begins to leave without responding to Horatio's questions, Horatio prevails upon it, "I charge thee, speak!" Clearly, he believes in its agency to do so; he is convinced that the ghost is "more than fantasy."
Horatio admits to Marcellus that he "might not this believe / Without the sensible and true avouch / Of mine own eyes." However, he has evidently been granted sufficient evidence from his own eyes; he is now convinced that the ghost of "the majesty of buried Denmark" is real.
Horatio shows he believes in ghosts in several ways. He tells Francisco and Bernardo that the ghost looks like King Hamlet, whose death has recently occurred. So it is obvious that he admits he sees something. Then, he tries to get the ghost to talk and seems genuinely frightened at its appearance. Finally, he suggests that they must find Prince Hamlet to see the ghost, in hopes that the ghost may speak to his still-living son. This begins one of the most common motifs of the play: reality or deception. The question becomes, "Is this really the ghost of King Hamlet or some evil spirit bent on destroying Denmark? "
After the ghost's appearance, Horatio exclaims, "Before my God, I might not this believe / Without the sensible and true avouch / Of mine own eyes." In other words, he had to see the apparition to believe in it.
When the ghost appears again, Horatio addresses the spirit and tries to get him to speak. The ghost, however, does not respond and exits the scene again. At this point, Horatio, Bernardo, and Marcellus decide that they must tell Hamlet what they have encountered that evening.