Howe, though, will keep believing a great escape remains possible ahead of next weekend’s season finale at Everton.

“The key thing is while there’s still hope we have to believe, until such time as it is mathematically impossible,” said Howe.

“You never know in football, crazy things happen.

“That’s my immediate thought, but very disappointed today. A big chunk of our destiny and our future was still sort of in our hands. It feels like that has slipped away a little bit.

“I am deeply disappointed, hurt, loads of different emotions are going through my body right now.”

Bournemouth, seeking to secure a much-needed first league double over their near neighbours, had the better of the opening period but struggled to create.

They were made to pay when former Cherries striker Ings claimed his 21st top-flight goal of the season four minutes before half-time.

In-form Ings then saw a second-half penalty saved by goalkeeper Aaron Ramsdale before the hosts thought they had snatched a late leveller through academy graduate Surridge’s low drive.

Their joy was swiftly cut short by an offside call against Callum Wilson made at Stockley Park, before Adams made matters worse by adding the visitors’ second.

Howe said: “It was a very emotional game in many ways. The lads gave everything to the match.

“I thought at times we played very well, created a number of chances but ultimately our finishing and the last part was just missing, the break of the ball was slightly difficult for us.

“We’re disappointed with the first goal we conceded because until that time they hadn’t really threatened our goal. And I think the second is probably as a result of the VAR decision at the end.”

Bournemouth must overhaul Watford, who appear set to replace manager Nigel Pearson, and hold off 19th-placed Aston Villa to prolong their Premier League stay to a sixth season.

Howe had little interest in discussing the managerial situation at Vicarage Road.

“I have no opinion on it, to be honest. I only care about my team,” he said.

Victory for Southampton ensured a small piece of history in their remarkable recovery from last October’s 9-0 humiliation at the hands of Leicester.

Saints have now enjoyed their most fruitful top-flight campaign on the road, registering a club-record 31 points away from St Mary’s to surpass the 28 achieved in 1983-84.

Manager Ralph Hasenhuttl said: “We do a much better job now with the ball, more quality creating chances, we are more flexible now.

“This is the key for being a better Premier League team. If you stop working hard, stop developing, we will be back to where we were at the beginning of the season. We were really hungry to be successful and it’s the reason we are 13th.”

Speaking about the game, he added: “The only thing that has disappointed me was that it was so long not finalising the game. If we had scored the second goal earlier, the game is done for us.”