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A few questions:

1)How does the wikipedia definition cope with infinity and - infinity?

2) Which definition is best to work with?

3) In his book Rudin states the result that if x is bigger than limsup, then it is an eventual upper bound of sequence. Bearing in mind he uses the first definition, he goes for a contradiction and says that if x is less than the sequence for infinitely many n, there exists a subsequential limit y exceeding x, contary to x being an upper bound. How he comes up with this y is mystery to me. Wonderful book but sometimes leaves out 'obvious steps'.