Spurs host Everton on Wednesday night in a game postponed from the opening day of the season following the London riots, and a win will take them level with Manchester United and just three points behind leaders Manchester City.
It is a far cry from when Tottenham got their campaign under way with a 3-0 defeat at Old Trafford and a 5-1 home loss to a rampant Manchester City.
Since then Spurs have been beaten only once, away to Stoke, and now lead London's challenge at the top of the table as they prepare to head to the Etihad Stadium on January 22.
"We have improved this year, we are stronger. It is just that belief," Redknapp said. "We have to believe we can do it and have that belief that we can keep challenging.
"I am not going to say 'We are going to win the league' as it is going to be very difficult - just look at Man United's squad and City's squad, they have fantastic players and can make changes everywhere. Their forward players are incredible.
"Fergie [United boss Sir Alex Ferguson] will look at it and say 'They are going well, they are playing well'. He is not disrespectful to write us off completely.
"It will be difficult for us to finish above them or Man City, but you just never know do you?
"If we got Champions League football again we would be delighted here. That is what we really wanted to do this year, anything above would be great."
Redknapp insists he was never short of confidence that things would improve once the likes of midfielder Scott Parker, goalkeeper Brad Friedel and on-loan Manchester City forward Emmanuel Adebayor were brought in.
"We were one or two players short and we had the [transfer rumour] situation with Luka [Modric]. There was a feeling around the club that it was a difficult period, that first couple of weeks," Redknapp said.
"But once we got that out of the way with the international break we were up and running. I knew that once one or two came in that we would push on and we did.
"The belief and the arrival of a couple of players in through the door made a difference to everybody."
With the transfer rumour mill now in full swing, Redknapp has found himself linked with moves for Marseille striker Loic Remy as well as Blackburn pair Christopher Samba and Junior Hoilett.
However, he insisted he is not "desperate" to sign any fresh faces this month just for the sake of it.
"We have good players like Niko [Kranjcar] and [Steven] Pienaar who are not getting a game, so to take another player and not have him playing is a waste of time," Redknapp added.
"The January window drives you mad. You get phone calls from agents without a clue, offering you useless players.
"If you do happen to answer by mistake, you would say 'We are in the Premier League, not the Southern League. You are ringing the wrong manager'. It is unreal."
This summer's civil unrest started just yards from White Hart Lane, which came after peaceful protests about the death of 29-year-old local man Mark Duggan and spread to other cities.
Redknapp added: "None of us could believe what was happening in London. It was scary everywhere and then spread to other parts of the country. It was a pretty horrific time."
