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Sheffield United boss Chris Wilder praised his players for "massively overachieving" as the Blades' hopes of securing European football next season were all but ended following defeat by Everton.
Richarlison's header early in the second half secured a victory for the Toffees and means Wilder's side cannot finish higher than eighth with one game remaining.
The Blades were favourites to make an immediate return to the Championship but have surpassed expectations with a superb season in which they are guaranteed to finish inside the top 10.
"The club and the players have overachieved massively," Wilder told BBC Sport. "To have had any talk of jumping into the top six means the players deserve an enormous amount of credit.
"We are delighted as staff we have got the maximum out of the players. They have left everything out there yet again.
"Yes they lacked that bit of quality in the final third but top 10 guaranteed and another season in the Premier League - that is what we should be reflecting on."
Wolves' win against Crystal Palace later on Monday effectively ended Sheffield United's slim hopes of playing European football next season, barring a hugely unlikely set of circumstances.
Richarlison's goal just after the break brought the game to life after a dire first half in which Dominic Calvert-Lewin had the only real chance, but his header hit the post.
David McGoldrick was denied a immediate response for Sheffield United when his low drive inside the box was blocked, but the Blades looked distinctly off key throughout.
Instead it was Everton who looked the most likely to add to their lead as Calvert-Lewin brought a save out of Dean Henderson before Andre Gomes drove just wide of the post in the closing stages.
The win for Everton moved them above Southampton and into 11th.
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"The spirit was really good," said Everton boss Carlo Ancelotti. "It is not a surprise, the players know really well what our supporters ask for on the pitch - to have a good spirit and fight and we showed that today.
"We had opportunities on the counter-attack, I am really pleased with the performance. We are at the end of the season but this fantastic spirit can be a good sign for the future."
Reaction to Sheffield United's defeat to Everton, plus the rest of Monday's Premier League actionAre Sheffield Utd the Premier League's best newcomers?Blades' impressive campaign starting to take its toll
Wilder was hugely critical of his players after the 2-0 defeat by Leicester last time out, calling them League One and Championship players playing in the Premier League.
The 52-year-old is known for his tough love approach to management - an attitude that has helped his squad play beyond their perceived individual ability this season.
Wilder clearly expected a positive reaction as he named the same side that started in that Foxes defeat. There was plenty of huff and puff from his players against the Toffees but they lacked quality in the final third.
There was also a distinct lack of urgency, with the toils of a long but hugely impressive first season back in the Premier League looking like it is starting to take its toll.
They pressed in the final few minutes as Everton sat back but finished the game without mustering a shot on target.
Regardless, Sheffield United have one game remaining of a superb season and they can be proud of their efforts.
Richarlison underlines importance to Toffees
This was a much needed win for Everton with their season having looked in danger of fizzling out in disappointing fashion.
A run of two wins and a draw after the restart was followed by a four-game winless run that ended their hopes of a top-10 finish.
Carlo Ancelotti approached this game with one eye on next season, assessing his options from the club's youth team by handing promising 18-year-old defender Jarrad Branthwaite his first Premier League start while 19-year-old forward Ellis Simms was included on the bench.
But whatever changes Ancelotti decides to make to his squad over the summer, this game underlined the importance of Richarlison to the Toffees' aspirations next season.
He was one of the few players for either side looking to make something happen in a dull first half with some determined runs, while it was his goal - a superbly guided header after a clever run - that finally provided the spark the game needed.
Richarlison has scored 15 goals so far this season - 14 of those coming in the Premier League - an impressive return in a difficult campaign for Everton in which they had looked in danger of being involved in a relegation battle before Ancelotti's arrival in December.
A full pre-season working with the Italian and the Brazilian has a real chance of developing into a 20-goals-a-season forward next term.
Man of the match - Richarlison (Everton)
A rare blank for the Blades - the stats
This was Everton's first away league win against Sheffield United since October 1972, ending a run of eight successive visits to Bramall Lane without victory.Sheffield United have lost four of their nine Premier League matches since the restart (W3 D2), as many as they had in their previous 21 before the break (W9 D8 L4).All five of Everton's away wins in the Premier League this season have been by a one-goal margin. They're the only side in the division this season to have had 100% of their away wins by a single goal.Sheffield United failed to register a shot on target in a home league game for the first time since August 2016, in a 0-3 defeat to Southend in League One.Everton have kept two clean sheets in their last four Premier League away games, having conceded in 14 consecutive games on the road previously.Everton have scored 14 headed goals in the Premier League this season - only Liverpool have netted more (16). It's the Toffees most headed goals in a single campaign since 2010-11 (also 14)Richarlison has scored 26 Premier League goals since joining Everton - among Brazilian players in the top five European leagues, only Neymar (28) has more in that time.All three of Gylfi Sigurdsson's assists in the Premier League for Everton this season have been for headed goals.What next?
Sheffield United complete their Premier League campaign with an away trip to Southampton on Sunday, 26 July (16:00 BST), while Everton host Bournemouth at the same time.
