Ewan - If your quote was treble the previous cleaner's price, I doubt you would get the job straightaway - they'd look for another cleaner first.
However, I've recently taken over a dozen jobs from a retired cleaner and all my quotes were one and a half to double his old price. I got a few raised eyebrows, but everyone accepted the quote.
I think most customers know when the've been 'on a good thing' for too long and are willing to pay a 'proper' rate for the job.
If they then see that the job is being done as well as, or better than, the last cleaner, they won't rebel when it comes time for another price increase (which could be after 4/5 cleans).
Mike