Walcott maintains Arsenal must believe in their ability to finally end their quest for silverware.
"It was tight, but we have deserved it [third place], for the past few weeks anyway," Walcott said.
"A couple of new additions may come, but we have to push on as it has been so long without a trophy now.
"We just need to start believing a bit more.
"The belief came in September that we really wanted it. If we can do that right from the start, keep people fit, then you never know where it will take us next year."