These are all taken with my travel zoom lens which is 28-300mm on my Full Frame Canon. Obviously the optics are compromised, but the correction is pretty good in Lightroom and Photoshop now, it's not really worth worrying about.
The Sony I listed is the previous model, the RX100 III is the new one but it's ?650-700 (superb camera though all the RX range is very good). Canon make great compacts too, I have the S110 which is excellent and very pocketable, and the S120 improves on that with most things.
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