Schnyder's reward will be a semi-final against third seed Jelena Jankovic, at the Italian Open.
Serena Williams, winner of eight Grand Slam tennis titles, says she has been inspired by her fellow American, boxing great Muhammad Ali.
Serena Williams powered into the fourth round of the Australian Open on Saturday with a 6-2, 6-1 demolition of Dayana Yastremska then consoled the Ukrainian teenager after leaving her in tears.
Serena Williams finished her last U.S. Open tune-up in style by blowing away Serbian Ana Ivanovic 6-4, 6-1 to win her first title at the Western and Southern Open in Cincinnati on Sunday.
Six-time Miami champion Serena Williams and five-time runner-up Maria Sharapova remained on another Sony Open collision course after each recorded fourth-round wins on Monday.
Serena Williams fought back from the brink of defeat to oust fourth seed Maria Sharapova 2-6, 7-5, 8-6.
Photos from the matches played at the Miami Open on Monday
Top seed Serena Williams won the final eight games to rout Andrea Petkovic in the semi-final at the Stanford Classic in California on Saturday.
The only former champion in the women's draw, Serena Williams, was ousted in the second round of the French Open.
Serena Williams's health and gamesmanship were the subjects on everyone's lips at the French Open.
Serena Williams, who won the Australian and Wimbledon Grand Slam titles last year, was voted the WTA Tour 2009 Player of the Year for a second successive season, at a ceremony in Miami on Wednesday.
Rafael Nadal made short work of his great rival Roger Federer to win the Italian Open, his sixth trophy of the year, 6-1, 6-3 on Sunday.
The former world number one struggled to a 7-5, 6-3 victory over Spain's Maria Sanchez-Lorenzo in the second round of the Rome Masters.
Serena Williams has still not kicked post Wimbledon jet lag, but she still managed to beat Chanelle Scheepers 6-4, 6-0 in the quarter-finals of the Stanford Classic on Friday.
Serena Williams battled her way into the Australian Open semi-finals with a 3-6, 6-2, 8-6 win over Israeli Shahar Peer.
Serena Williams's motivating factor in her WTA Championships triumph on Sunday was to save her own sanity, the American said after beating Maria Sharapova in the final of the season-ending tournament.
Fatigue and a busy schedule was not going to get in the way of Serena Williams returning to the place she says launched her comeback and gave her the confidence to believe she could win another Grand Slam.
American Serena Williams reached the semi-finals at the WTA Tour Championships in Istanbul on Thursday with what is fast becoming a routine defeat of World No 1 Victoria Azarenka.
Also making it to the last eight were Svetlana Kuznetsova and Czech teenager Nicole Vaidisova.
Williams has not played competitively for nearly eight months after undergoing a knee operation but had hoped to make her comeback at the $600,000 event in Doha.
Serena Williams showed tennis was not just the domain of teenagers as she climbed back to the top of the world rankings on Friday, becoming the oldest female to hold the coveted position at the age of 31.
Serena Williams marched into the second round of the US Open on Tuesday and despite being seeded only fourth at Flushing Meadows, the American is in no doubt she is the world's best player.
Serena and Venus Williams added another chapter to what the world number one claims to be greatest story in tennis when she outslugged her sister 6-2, 1-6, 6-3 on Tuesday to reach the semi-finals of the US Open.
Serena Williams trounced Maria Kirilenko at the Australian Open on Monday, setting up an all-American quarter-final with her would-be successor Sloane Stephens after Andy Murray had put flagging Frenchman Gilles Simon out of his misery.
"I'd like to thank all of the fans who have been so supportive. I'll be back... better than ever," said the world number three.
World No 1 Williams, 33, could not practise on Friday, still suffering from an illness that left her coughing and spluttering during her three-set semi-final win over Swiss Timea Bacsinszky.
Tournament favourite Serena Williams withdrew from the Italian Open on Saturday with a lower back injury, the WTA said.
Victoria Azarenka and Serena Williams will square off for the Madrid Open title after the current and former World No. 1's swept into the final with a minimum of fuss on Saturday.
Henin and Williams have met eight times in their careers but Saturday's clash will be the first since the muscular American won a semi-final meeting with the Belgian at Wimbledon nearly four years ago.
Serena Williams revived memories of her two Australian Open titles by coming from a set and a break down to beat Russian fifth seed Nadia Petrova 4-6, 7-5, 6-3 and book a place in the fourth round on Friday.
Serena Williams reached the Family Circle Cup semi-finals on Friday by beating number two seed Maria Sharapova 7-5, 4-6, 6-1 while Vera Zvonareva upset top seed Jelena Jankovic 6-2, 3-6, 6-2. Sharapova, who won only nine of 35 points in a final set during which Williams made only one unforced error, lost to the American for the fifth time in seven career matches.
Germany's Angelique Kerber, who will try to prevent Serena Williams winning an eighth Wimbledon title on Saturday, never went away but she too is making a comeback of sorts.
Both will play their second-round singles matches on Thursday.
Serena Williams retained her Sony Ericsson Open title with a 6-1, 5-7, 6-3 win over fourth seed Jelena Jankovic in windy conditions on Saturday. The eighth-seeded American, who is now tied with Germany's Steffi Graf for the record of five Miami titles, triumphed on her eighth match point with an overhead smash.
Ruthless Serena Williams underlined her dominance at the top of women's tennis on Sunday, completing her year in style with a 6-2, 7-6 defeat of big sister Venus in the final of the WTA Championships.
Serena Williams produced 20 aces and blasted her way into the quarter-finals of the Sony Ericsson Open with a 7-5, 6-3 win over Australian Samantha Stosur on Monday.
A summary of Wednesday's action at the US Open on Day 3.