Tottenham are not making things easy in their quest for a top four Premier League finish. On a madcap afternoon at Bournemouth, they played for 42 minutes with nine men, after seeing Son Heung-min and the substitute Juan Foyth sent off, while they were fortunate not to lose Eric Dier to another red card. Their rap sheet also featured five yellow cards.

Mauricio Pochettino’s depleted team clung on until stoppage-time and it had to be said that Bournemouth were strangely toothless opponents. But the home team, who have struggled so sorely at home of late, found the killer blow when Nathan Aké showed more desire than anybody around him to head from Ryan Fraser’s corner.

The upshot for Spurs was that unless results involving the teams immediately below them go in their favour on Sunday, they will need a result – possibly a win – at home to Everton on the final day of the season next Sunday.

Spurs’ poor form goes on. They have now lost seven of their last 11 league games and this was not the preparation they had in mind for the second leg of the Champions League semi-final at Ajax on Wednesday, in which they trail 1-0.

Pochettino’s players were the architects of their own downfall, with Son getting himself into trouble when he momentarily lost his cool on 43 minutes to push Jefferson Lerma and Foyth leaping into a poor challenge on Jack Simpson almost immediately after his introduction as a half-time substitute.

Bournemouth played like a mid-table team at the end of their season but they could enjoy the finale when Aké powered through a crowded penalty area to strike the decisive blow. The irony that the concession followed a set-piece, when Spurs had the man-power to defend properly, was not lost on Pochettino.

Spurs finished the first half with 10 men but their complement could and should have been shorter. Dier skated on thin ice from the moment that he was cautioned for taking out Fraser as the winger led a break and there was outrage from the home support when he slid back to trip Joshua King on another Bournemouth counter in the 34th minute.

TV replays showed that Dier did not get anything on the ball and he then raised his leg a little to make sure that he halted King. The entire Bournemouth team moved towards Craig Pawson to demand Dier’s dismissal and when he allowed play to continue, they were almost caught cold when Spurs broke through Dele Alli. Pawson had the whistle to his lips before appearing to think better.

The shock was palpable. Dier got away with one and he would get away with another rash challenge on 42 minutes when he jumped into Callum Wilson inside the area. Dier got none of the ball and all of the man, and it was a mystery as to how Bournemouth did not get a penalty. Pochettino withdrew Dier at the interval because he had become a danger to his team.

Pawson’s straight red card for Son felt a little harsh. That said, Son did give him a decision to make because he raised his hands and shoved Lerma, having felt the frustration after conceding a free-kick to Fraser. Lerma went down dramatically. Son ought to know better.

Pochettino replaced Toby Alderweireld – who was on a yellow card – with Foyth at half-time and the young Argentinian let him down badly. He had been on the field for a little over two minutes when he over-ran the ball and launched into a high tackle on Simpson. The only thing to say in his defence was it was not two-footed. Pawson had no other option.

Prior to the dismissals, the central figure had been Mark Travers, the 19-year-old Bournemouth debutant, who became the first teenage goalkeeper to start a Premier League game since Joe Hart for Manchester City in October 2006.

Travers’s afternoon almost started badly when Lucas Moura spotted him off his line and nearly beat him from the halfway line with an excellent effort. But Travers quickly got his bearings and he made a clutch of saves in the first half hour or so, the best of which was the flying tip over to keep out an Alli header.

Travers could look back on a more routine stop to deny Alli; a tip over from Lucas and a smart block to thwart the Brazilian again. Son also blew a good chance on 31 minutes when he ran around Steve Cook only to lift his shot high. Perhaps, that was the beginning of his irritation.

It was a curious performance from Bournemouth. When Spurs had 11 men, they allowed them space, particularly in between the lines for Christian Eriksen and Alli and, when they held the numerical advantage, they hardly laid siege to the Spurs goal.

King had been denied by a Hugo Lloris block on 38 minutes while Simpson headed straight at the goalkeeper in first-half stoppage-time. Thereafter, Jordan Ibe worked Lloris from distance and the substitute, Lys Mousset, blew a free header. Aké showed them how at the death.