Caroline Wozniacki has spent most of her life proving herself.

Initially to people who doubted she could make it as a professional and subsequently to those who said she did not deserve to be world No.1, and that her game lacked the aggression to make her a grand slam winner.

The Dane is one of the most like­able players on tour, always generous with her time and polite, even in the face of some of the most intrusive questions regarding her private life. Underneath the broad smile, though, is a steel which took her to the top of the world rankings and which now, long after she had been written off as a potential grand slam tournament winner, to within one victory of reaching a second US Open final.

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"I have proved people wrong so many times," said Wozniacki, who will be the big favourite today when she plays Peng Shuai of China in the semi-finals. "I was told when I was younger there is no chance I will make the top 100, top 50, top 30. Every time I have proven them wrong. It's kind of nice."

At the age of 24, Wozniacki seems to have been on tour for ever and indeed, this is her eighth US Open. When she reached the final in 2009 as a 19-year-old, it seemed as if Woz­niacki would go on to big things. But although she topped the world rankings at the end of 2010 and 2011 and was the best defensive player around, Wozniacki found herself over­powered and her lack of a real weapon left her wanting.

Switching among a number of coaches and her father, Piotr, did not help and Wozniacki had to work hard to stay in the top 10. Her relationship with golfer Rory McIlroy led to more headlines than her tennis did for a while but, since their widely-publicised break-up, Wozniacki has clearly dedicated herself to the game and the results have followed.

In her 6-0, 6-1 victory over Italy's Sara Errani on Tuesday night, Woz­niacki did not miss a single return; she is moving better than ever and there seems to be a bit more pop on her serve. "My game has definitely evolved," she said. "I know my game much better and I have been in these big matches before. I keep putting things in to my game. I try to have stronger serves, better returns, on the first few points."

Her match with Peng is her first grand slam semi-final in exactly three years, but Wozniacki said memories of her run to the final in 2009 were even fresher in her mind. "I have to say the final feels closer than the semi-finals of 2011," she said.

"It's weird. I think just because it was such a huge experience for me and it was my first really deep run in a grand slam. I'm just so happy to be back in this stage and having another match out there."

One of the most interesting features of the summer has been the way that Woz­niacki's friendship with Serena Williams appears to have rubbed off on her tennis. The pair spent beach-time together and Woz­niacki pushed the world No.1 to three sets both times when they met in pre-US Open tournaments.

Williams, who was scheduled to play her quarter-final against Italy's Flavia Pennetta last night, is in the other half of the draw and Wozniacki joked that the two had discussed what might happen if they meet again. "I have had a great summer and I told Serena I'm pretty tired of her," Wozniacki said, flashing a smile. "Twice she beat me in three sets. I said, 'Can you just get out of my way?' We just laugh about it."

As popular a champion as she would doubtless be, there remains one more match before she gets to the final. Wozniacki said she has full respect for Peng, the world No.39, who hits with two hands on both sides and who is into a first grand slam semi-final. "It's going to be a hard one but it's going to fun," Wozniacki said. "I feel confident and comfortable out there. I have been playing well these last few months."

Peng's story is a pretty extraordinary one, having had heart surgery when she was 12 and returned a year later with doctors unsure whether she would be able to carry on. "Everybody was saying to me all week, maybe you have a chance to get quarters, so I was really happy to be in the quarters," she said. "Of course it is exciting, it is a surprise, but it is not like I feel I cannot [win again]."