Saints lock Day: We were our own worst enemy
The lock believes individual errors were the key to his team's undoing in their 43-32 loss, which was a ninth in 10 matches for Jim Mallinder's men.
Ospreys, who had not won away in Europe's top tier tournament since 2009, scored six tries at Franklin's Gardens.
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Hide AdSaints did fight back in the final 20 minutes, and they scored five tries of their own.
But it was too little, too late as the soul-searching continued at the Gardens.
"We need to look at ourselves," Day said. "There's no point looking for a quick fix.
"I missed a tackle for a try and everyone has got something to work on.
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Hide Ad"We knew the Ospreys were a good team - they've got British Lions playing for them and they opened up a big lead.
"We were our own worst enemy and we're very disappointed.
"It wasn't a lack of effort or physicality - it was mistakes.
"I wasn't trying to miss a tackle on their full-back (Dan Evans), who had a great game, and he put seven points on the board.
"It was two teams wanting to win and we weren't good enough."