The north London club currently occupy top spot but lost their last league outing against reigning champions Manchester United and face another stern test in the form of high-flying Southampton on Saturday.
Szczesny, though, says Arsenal's form over the last six months is proof they can end their wait for silverware, whether manager Arsene Wenger brings in reinforcements in January or not.
"I don't want to say nobody can challenge us for a trophy but if we are at our best we will win a trophy and that is all that matters," the goalkeeper told reporters.
I wouldn't say we have proved everyone wrong and passed all the tests. I will say that when we get our hands on a trophy in May.
"I think what is important is what we believe in and we believe that we are good enough with the players we have got and the work we are doing to win the Premier League.
"We have shown that to people over the last six months or so. We know that if we just carry on doing the same thing we will show people that this team and these players are good enough to win a trophy without any extra additions or extra work that people are talking about. We certainly believe we can do it.
"We need to make sure we continue doing what we have been doing recently and keep getting the results."
