How does Birece convince you that he has not escaped though he is writing as if he had?"An Occurrence at Owl Creek Bridge" by Ambrose Bierce

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,

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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.