The phrase "There are no smallest numbers in every subset", unfortunately, is ambiguous in English. I hate when people say, "Every group of people does not have a leader" when they mean "Not every group of people has a leader". So, if by "Every subset does not have the smallest element" you mean what this phrase should mean, i.e., that there is not a single subset that has the smallest element, then this is not the negation of "Every subset has the smallest element".
The problem directs you to first write the original statement in quantifiers. Then negating it would be easy.