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For what it's worth we also know that f is bounded below and that there exists a y ∈ X such that f(y) equals the infimum of the range of f. ( these were the first two parts of the problem which I have already successfully proven ).

I literally can't think of any kind of compact metric space which satisfies these requirements, it must be pretty far out there. My approach to the first two parts was by contradiction: assuming the negation of the consequent and then finding an open cover which had no finite sub-cover, hence showing X wasn't compact. But this doesn't seem to work for this last part, and I can't find a good approach to it. Thanks.