“He resisted going into any program for a long, long time,” Lucille Ball and Desi Arnaz’s daughter Lucie Arnaz tells people

Desi Arnaz reached sobriety toward the end of his life, a proud and emotional moment for his children.

In this week’s issue of PEOPLE, Arnaz and Lucille Ball’s daughter Lucie Arnaz, 70, reflects on her famous parents’ relationship ups and downs, including how his drinking sometimes got in the way. Lucie, who appears in the new documentary Lucy and Desi, says Arnaz got sober with the help of her younger brother Desi Jr.

“He resisted going into AA or any program for a long, long time. We don’t air our dirty laundry in front of other people; he hated that part. But he finally did after his wife Edy died,” she tells PEOPLE. “He was just so upset that my brother convinced him, I think, ‘Come on, Dad. You can do this.’ And he did it. He stopped drinking.”

“I was right next to him when he stood up and said, ‘My name is Desi and I’m an alcoholic.’ It was the proudest moment of my life to sit next to him and hear him do that,” says Lucie. “It was like, ‘Okay, I don’t care what you do after this. That was the best thing you’ve ever done.’ Because that was hard for him. It’s hard for everybody, but for him especially it was really hard.”

She adds, “It was a big hurdle for him to do that, and I was very proud of him. That takes incredible amount of courage.”

Sadly, Arnaz found out he had lung cancer shortly after getting sober. He died a year later in December 1986 at age 69.

Though Ball and Arnaz’s marriage ended in a bitter divorce back in 1960, their “unconditional love for each other” lasted and “never went away,” Lucie recalls: “There was forgiveness. There was understanding. There was an appreciation for that other person and what you got from that other person, regardless of the stuff that hurt that never went away. And that was beautiful and that, in and of itself, was a wonderful thing to leave us.”

Lucy and Desi debuts March 4 on Amazon Prime Video.