The England midfielder is delighted with the Gunners' finish to the campaign but says that the club must hit the ground running next term if they are to compete for the title
The Gunners finished this term in fine form, securing fourth place and Champions League qualification, after picking up 26 points out of a possible 30 from their last 10 games.
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He told the club's official website: "The games earlier on don't decide our season but we have got to think that we have to win them, because that is what champions do.
"The likes of Manchester United go away and win games and leave you thinking: 'they keep winning and winning' and, at the end of the season, the points tally shows that.
"That is what we have to do [to challenge for the title] and we also have to improve our record against the big teams. It has not been great this season and, if you want to become champions, you have to beat those sides."
Wilshere will miss England's friendlies against the Republic of Ireland and Brazil to undergo a minor operation on his right ankle, but is expected to be fit for Arsenal's pre-season Asian tour.