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added statement and proof that sequentially compact metric spaces are equivalently compact metric spaces
What do you mean by “being compact is strictly stronger than the condition of beiing sequentially compact”? – For a general topological space both properties are independent, see Wikipedia for an example of a compact space that is not sequentially compact.
Thanks for catching this. Fixed.
Added comment on generalization to first-countable spaces of one implication.
Thanks. But let’s make that more explicit: countably compact metric spaces are equivalently compact metric spaces
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