On the contrary, the reader is not convinced that Farquhar is dead. Why continue reading if this is the case?
In his essay, "'Something uncanny': The Dream Structure in Ambrose Bierce's 'An Occurrence at Owl Creek Bridge," Peter Stoicheff writes,
Peyton Farquhar's death at the end is a surprise,so carried away are we by his escape, yet it seems somehow presaged by the very description that keeps it, until the story's last paragraph, obscure and unanticipated.
So, while there may be suggestions of Farquhar's death, they are ambiguous enough with the surreptitious switch to objective, rather than omniscient narrator, that the reader accepts the narrative of Farquhar's "awaken[ing]--ages later, it seemed to him." As, in reality, Farquhar's experience is that of the dying in which time--forever a state of the mind anyway--divides infinitesimally into the twenty-four hours of his escape. The metaphor of the pendulum conveys how Farquar's mind "swung through unthinkable arcs of oscillation." That is, not only did time slow down, but, as Stoicheff contends, it "opens it from the inside." In retrospect at the story's conclusion, the reader realizes that the metaphor of the pendulum is the comparison of Farquhar's body swinging as he is hanged, but Bierce is able to create the dream enough that the reader gives it credibility. The amazing parallels between what Farquhar dreams in that turning of time inside out against the reality are what lend verisimilitude to his dream. And, it is not until the final paragraph, the surprise ending, that the reader is shaken from this dream.
Bierce is careful to point out that Farquhar is feeling "pain," that he is "suffocating," that there is "a sharp pain in his wrist," and that "his neck ached horribly; his brain was on fire" (paragraphs 18, 19). Also, Farquhar’s "visible world" is wheeling "slowly round" (paragraph 21). These descriptions realistically describe the circumstances of hanging that he is experiencing, even though he is also dreaming that he is making his escape. It is this wonderful dichotomy and Bierce's own unique style of writing that has made him a favorite of many contemporary students.