Arsene Wenger's position as manager of Arsenal has come under scrutiny over the last year, with his decision to sign a two-year extension at the club in the summer met by disapproval from many Gunners supporters.
"They've been in there the last few years so now I think they have to make sure they win something to show the progression", Cech told Sky Sports.
"When we played against Tottenham, usually we beat them".
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Despite their rapid improvement under Pochettino, Spurs are still in search of their first trophy since their League Cup triumph over Chelsea back in 2008 and Cech believes it's high time talk of a north London powershift was backed up with silverware.
Pochettino's Spurs have challenged for the league title over the last two seasons, while their recent victories over Real Madrid and Borussia Dortmund have placed them among the top clubs in the Champions League.
"The big factor was the appointment of Mauricio Pochettino", Cech added.
Arsenal lifted the FA Cup in May, their third time winning the prestigious trophy in four years.
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There have been some mitigating circumstances, with the Chilean arriving back late from global duty during the summer, and the German hampered by injury, but here we are heading for mid-November and between them they have just two goals and four assists.
Most of Mauricio Pochettino's star players will have had breaks from football during the global fixtures, with the likes of Harry Kane, Dele Alli, Jan Vertonghen and Hugo Lloris all picking up injuries that kept them out of contention for their countries.
"They are an important part of the team", said Cech.
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