What does the narrator do every night that he believes is proof of his sanity?

Like a number of Poe's unreliable narrators, the narrator of "The Tell-Tale Heart" is at great pains to convince us that he's completely sane, despite the fact that his behavior would suggest otherwise. Right off the bat, the narrator admits to some kind of nervous disorder, but he quickly goes on to reassure us that this doesn't mean that he's mad. He sets out to prove his sanity by telling the story as calmly as possible. Surely a madman wouldn't be able to do that?
The narrator relates the meticulous preparations he made leading up to the old man's murder. Each night, around midnight, he'd slowly open the door to the old man's bedroom. Then he'd gently open the shutter on a lantern to cast a small beam of light on the old man's evil eye, the one that really creeps out the narrator. But because the evil eye remained closed for seven whole nights, the narrator was initially unable to carry out his dark and deadly deed. The narrator is once again trying to convince us of his sanity. He's effectively saying that if he really were a madman, then he would've killed the old man at the first opportunity, instead of showing such incredible patience over the course of a whole week.