The Gunners were in the hunt for both trophies last year, but crashed out of the Champions League in the quarter-finals and finished as runners-up to Manchester City in the Premier League.
Arsenal lost just two of their final 20 games across all competitions but came undone during a busy spell in April.
In the space of four days, they lost to Aston Villa, which left City two points clear with six games to go, and were defeated by Bayern Munich in Europe 3-2 on aggregate.
Arteta is determined to fight on two fronts this season and Arsenal are so far unbeaten across all competitions.
They host Paris Saint-Germain at the Emirates Stadium on Tuesday night and asked if his squad has the ability to balance competing on multiple fronts, Arteta said: “This is what we have to prove.
“Last year we almost went all the way. We came up short, so that is something we have to do better.
“We have to be involved in the cups much further too and that’s the next step we have to make as a team.
“We are going to find out, but we’re not thinking about (winning the Champions League) yet because we are still very, very far away and we have a difficult game on Tuesday night.”
Defender Jurrien Timber, who is due to keep his place at right-back against PSG in the absence of Ben White, believes Arsenal are strong enough to fight for multiple trophies.
“We have a good, deep squad that can achieve amazing things if we keep going like we are and like the team was already doing for the last two years,” he said.
“We’ve been through a lot in the last few weeks. Things happened in matches that we didn’t expect or could be prepared for, but the team showed again and again that we’re ready for everything that’s coming. We have to be if we want to achieve amazing things this season.”
