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It did seem very strange that the guy seemed to be happy to build on top of the existing carpet, but I’m pretty sure I must have just not been completely clear what I was asking at the time. He comes very highly recommended and had a 6 month waiting list so I’m sure it’s my confusion. So long as I know that there is a right way of doing it and can insist on it if it comes down to it, it takes a weight off my mind - thanks for confirming!
It's not just the right, but the only option.
If your builder was happy to leave the carpet down, fire them.