The protagonist of Oedipus Rex seems to be heading towards peace rather than horror. Elaborate, supporting your answer with examples.
I am afraid your original question didn't make sense, so I had to edit it and try to guess at what you were originally meaning. I hope I got the sense of the question right.
However, if this was your question, I am afraid I have to disagree with it. Throughout the play there is a brooding sense of the horror that the audience knows is going to befall Oedipus. This is part of the dramatic irony of the text. We, as the audience, already know what Oedipus sets himself to find out, and we can anticipate how this knowledge is going to impact him. Thus it is that we watch his detective work through our fingers, dreading the moment when all is revealed.
Unfortunately, when Oedipus is "successful" in his investigations, his act of self-multilation, which is deeply symbolic, clearly indicates the horror of this self-knowledge. Consider some of his last words that clearly indicate the terror and sadness of his position:
And now what is left?
Images? Love? A greeting even,
Sweet to the senses? Is there anything?
Ah, no, friends: lead me away.
Lead me away from Thebes.
Lead the great wreck
And hell of Oedipus, whom the gods hate.
This play presents us with the desperately tragic fate of one central character whom it is clear that the gods have destined to suffer intensely and greatly. Therefore there is no sense of peace in this play, and no arrival of any "happy ending." We leave this play disturbed by the plight of the protagonist.