I'm 99% sure this story is by the same author as this other story I asked about recently. Like that story, I'm not sure if I read it here or on some other site.
A college age kid is walking by a bridge when he sees a woman his own age about to jump. He tries to talk her out of it, but she says she's too jaded and jaded to enjoy life anymore. He asks her to go on a date with him to delay her, basically, and they end up having a great time together.
She's still determined to end it after it's over, though, so he walks her back to the bridge and then walks to the other side to commit suicide himself. She gets pissed at him but he says he was already intending to kill himself when he found her. She's too jaded to live, but he's too scared and timid to do anything at all and enjoy life. She brought him out of his shell for a night, but he doesn't want to go back to the way it was, so he's going to kill himself.
The story ends with the girl asking the guy out on a date, and basically they're going to take life one day at a time.