Michael Phelps’ story finished with a twist, as the best ones always do. He lost the last solo race of his career, the 100m butterfly final, to Joseph Schooling, a 21-year-old from Singapore. Phelps wasn’t too cut up about it. He has 22 gold medals, but this was Schooling’s, and Singapore’s, very first. Phelps felt he could spare it. And more than that, he found a satisfaction in defeat that seemed almost to matter more to him than another gold medal.

“I’m not happy, obviously, nobody likes to lose,” he said. “But I’m proud of Joe.” He was in a reflective mood. “I wanted to change the sport of swimming,” he said. “With the people we have in the sport now I think you are seeing it.” He explained he wanted to teach kids “to believe in themselves, to not be afraid to know that the sky is the limit.” And that’s exactly what he has done.

“Just being beside him,” Schooling said, “walking alongside him and celebrating, I will cherish that for the rest of my life.” They have known each other for a long time. They first met in 2008, when the US team stopped off in Singapore for a training camp before the Beijing Games.

“They came to the country club that I trained at,” Schooling said. It was early in the morning, and he was working on an essay. “Everyone just rushed up and was like “it’s Michael Phelps! It’s Michael Phelps!’ and I really wanted a picture.”

Phelps obliged. Schooling looks adorably awkward in it. “It was very early in the morning,” he remembered, “and I was so shell shocked, I couldn’t really open my mouth.” Eight years later, Schooling’s winning time of 50.39sec broke the Olympic record Phelps set at those same Beijing Games.

Phelps, swimming in lane two after a slow semi-final, finished in 51.14sec, which left him in three-way tie for the silver medal along with his two old rivals Laszlo Cseh, who he’s been racing since 2004, and Chad le Clos, who beat him at London 2012. It was the first triple tie in Olympic history, and the three of them took to the podium together, all holding hands, Phelps, Le Clos, an Olympic champion, and Cseh, the world champion. It was a special moment, and, as Phelps said, “a lot of fun”.

Schooling was, understandably enough, utterly overcome. Phelps helped him through it. On their victory lap, Schooling turned to Phelps and said, “Dude this is crazy, out of this world, I don’t know how to feel right now.” Phelps smiled, and said “I know”.

They say you shouldn’t meet your heroes. Schooling beat his. “If it wasn’t for Michael, I don’t think I could have gotten to this point. I wanted to be like him as a kid,” Schooling said. “I think a lot of this is because of Michael. He is the reason why I wanted to be a better swimmer.”

He said he was “humbled”, that “I don’t think I’m anywhere close to these three guys right next to me, I just had a good day today.” It was better than good. Schooling has had to handle the enormous pressure of being the best prospect in his small country. The President of Singapore had come along to cheer him on. Schooling said he hoped his win would show “that even people from the smallest countries in the world can achieve a lot of things”.

Schooling isn’t the only swimmer here with a snap of Phelps in his family album. Katie Ledecky has one, which was taken when she was nine. So does Daiya Seto, the 22-year-old from Japan who won bronze in the 400m medley. Phelps is now racing against and alongside the generation he inspired to take up his sport. And that’s the very legacy he wants to create for himself.

“Being able to continue to change the sport is something I am looking forward to,” he said. “Daring kids to dream. That’s the only reason why I’m sitting here. I was a little kid with a dream.” He hinted at some grand plans for the sport, which he says he will reveal after he’s retired.

First Phelps has one more race left, in the medley relay. Schooling and le Clos both told him he should stay on for four more years. And, just as they did at London 2012, the two Dwyer twins, Conor and Spencer, stood behind the podium chanting “four more years! For more years!”

Phelps’ friend Ryan Lochte has already predicted that he will buckle, and be back again, “just because when you get to this stage, we thrive off that excitement of getting on those blocks and racing the seven best other swimmers in the world”.

Phelps wasn’t having it, not any more. “No. No. No. Nope.” he said, “I’m happy with how things finished.” After 24 years in swimming, he has found something he values as much as winning. Phelps is at peace, even in defeat.