What are you thinking of? Yeah, compressing the air at the base will increase its temperature, and then the pressure (and so temperature) will reduce as it gets

What are you thinking of? Yeah, compressing the air at the base will increase its temperature, and then the pressure (and so temperature) will reduce as it gets higher, but it’s still going to be at a higher pressure than the air outside and so can provide support and reaction mass. You might even be able to take advantage of this by chilling the compressed air before it goes into the beanstalk, so that it absorbs heat from the surrounding air on the way up. Or have separate pipes with one carrying methane to burn as fuel. Lots of options to evaluate.