Spurs had to work hard for victory on Saturday and needed a Harry Kane second-half penalty to end Middlesbrough's resistance.
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"Nine points is okay, it's three games but it's up to Chelsea now. It's up to us to keep pushing, win games and it's important for the Premier League too," said Tottenham boss Pochettino. "I think we have more chance to try to compete and fight and be one of the contenders for the Premier League. We'll see what happens, there are a lot of games left.”
Tottenham dropped points against Sunderland in their previous game and there was a danger of history repeating itself on Saturday.
Middlesbrough had already gone to Arsenal and Manchester City and returned with a point and Pochettino's side were left frustrated by Middlesbrough goalkeeper Victor Valdes' heroics.
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Finally the pressure told when Bernardo Espona hauled down Heung-Min Son for a 58th minute penalty that was converted by Kane.
It could have been more but Pochettino believes his young team are coming of age.
"We learned a lot from last season and the team is very competitive," he said. "It's good to put the standard higher. I am happy to put pressure on us and say we are a real contender and we are a team ready to win the title. I manage pressure very well, no problem."
Middlesbrough are looking over their shoulders at the relegation zone and their lack of goals could come back to haunt them.
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But manager Aitor Karanka believes his side are good enough to retain their top-flight status.
"We have shown we can compete against every team. The players have been amazing," he said. "I have trust in them. We have to think about ourselves rathe than other teams. We are good enough to stay up. The commitment they put in is amazing. If we keep playing with the same organisation and attitude we will be okay."
Valdes marred a fine display by clashing with Dele Alli and he could face FA action after appearing to elbow the Tottenham midfielder.
Karanka played down the incident and said: "I don't think it's serious. Alli finished the game."
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