"immediate proximity" is correct and usable in written English.
You can use it when referring to two or more things that are located extremely close to each other and/or within a very short distance from each other. For example: The two cities are located in immediate proximity to one another.
Worse still, Plymouth, with a population of a quarter of a million people, would be in immediate proximity.
"You're telling the people in the immediate proximity about the sharpshooting that will take place," he said.
Some of us have chosen a more immediate proximity, as we use our bodies to confront and disrupt corrupt state practices.
We are, after all, confined to the immediate proximity of our spinning globe on the outskirts of the Milky Way galaxy.
Despite his racing pedigree, and the immediate proximity of a successful racing stable, Cecil showed little interest in racing during his days at Canford school, Dorset.
The edgy atmosphere takes on a frisson of horror when her lover rings her mobile to tell her of his marriage next day: from that moment, our immediate proximity to L is appalling for involving us implicitly.
Immediate proximity played a crucial role in obesity, according to the report, the Effect of Fast Food Restaurants on Obesity, by researchers at the University of California, Berkeley, and Columbia University.
The framework contract will be available to a range of public sector organisations in East Sussex, Brighton and Hove, West Sussex and the immediate proximity over four years, according to a notice in the Official Journal of the European Union.
With no opponent in immediate proximity his left ankle appeared to give way on landing which meant he fell on his right leg, and stayed down clutching his knee.
How does it feel to wake up, night after night, in immediate proximity to a crazed Pomeranian or a Malamute or an Old English sheepdog stubbornly adhering to the once caressable jaw of the guy on the nearest pillow?
Unfortunately, Barry seems to have a destructive effect on the lives of anyone in his immediate proximity, and Tim soon finds himself without a girlfriend and in danger of destroying any chance he ever had of attaining promotion.
