Harry Redknapp undersold it last week, which is not an accusation that the Tottenham Hotspur manager can face too regularly. When he estimated Manchester City's transfer market spend this season, he put the figure at £130m or £140m. He had beefed it up to £160m by Tuesday, on the eve of Real Madrid's visit, and it was closer to the truth. The precise figure is £154.46m.

Redknapp has mentioned the Eastlands numbers with increasing regularity of late and the reason is increasingly obvious. Once again he sees City as his most direct rivals for a fourth-placed Premier League finish and there is little he likes more than to paint his team as the big-hearted underdogs, a role that they played with no little panache at White Hart Lane against the nine-times European champions.

Tottenham's thrilling Champions League run has ended at the quarter-final stage but a greater achievement beckons over the season's final weeks. If the north London team can reel in City and gate-crash Europe's elite competition again next season, it may come to be considered as more worthy than their feat in setting up this fixture against José Mourinho's swaggering superstars.

The challenge is afoot and, exactly like last season, it is daunting. Redknapp's team must play Arsenal, Chelsea and City, not to mention Liverpool, in their final seven league fixtures. The focus must switch from the heady European nights that have energised the season but Redknapp is confident. He treated this no-win second-leg tie against Real in isolation. It would not affect anything, he said. Yet his players must ensure that the exit does not knock them.

This was a night when gallows humour stayed the course. When Mourinho swapped Cristiano Ronaldo for Kaká, the home crowd asked "Who are ya?" and they also insisted that they were "Gonna win 6-1" after Heurelho Gomes's fumble from Ronaldo's spinning shot. Then they put forward a darkly ironic prediction: "Thursday nights, Channel 5 …"

You had to hand it to the supporters, who had seemed drunk on blind hope. In the fast and furious opening exchanges, they had roared their encouragement on each and every positive flash. They seemed determined not only to believe in the impossible but to do their bit to make the night memorable. Real do not come to town every day. How the Tottenham faithful hope to see their ilk again. Soon. The Europa League could feel like a letdown.

"If we could make the top four again it would be a great achievement," Redknapp said. "It's even harder this year because of the improvements Man City have made, but we've still got a big chance. It's got the makings of another great end to the season." Spurs are fifth, three points behind fourth-placed Manchester City, and have seven matches left – one more than City.

Redknapp was entitled to have a little moan about the penalty appeals that his team saw turned down in the first half. If Gareth Bale went down too softly in the third minute under Xabi Alonso's challenge, then the Real midfielder was fortunate to get away with his tackle on the outstanding Luka Modric moments later. Ditto Raúl Albiol on Roman Pavlyuchenko after 28 minutes. "I thought it was definitely a foul on Modric," Redknapp said.

Tottenham threw what they had at Real and had they scored at any point in the first half, the excitement levels might have soared off the scale. There was a breathless quality to their football in the first period, an urgency to the quest for the goal that might have fired the collective dream.

Bale pointed to an imaginary watch on his wrist when Mesut Ozil dawdled over the taking of a corner – it was the 36th minute – and outrage swept the stands when the fourth official indicated just one additional minute in the first half. What a pity that Gomes's blunder altered the mood so abruptly. "The keeper's been great for me," Redknapp said. "He's made a rare mistake but that's how it goes."

Redknapp has already outlined what he feels it will take for Tottenham to "have another real go" at finishing in the top four next season, namely "one or two signings that can make the difference". There is little doubt that his priority is a striker.

His selection against Real Madrid was the latest blow to Jermain Defoe. Redknapp never sounds like too big a fan of Pavlyuchenko and he held his head when the Russian wasted two clearcut chances but he nevertheless started him ahead of Defoe, who might wonder whether he has a future at White Hart Lane.

Redknapp is happy to rely on what he has in the battle to leapfrog City or even Chelsea and, when the night fell flat on Gomes's horror moment, the search for portents was on. One thing stood out. Redknapp's players did not stop until the very last. It was a statement of intent for the business ahead.