MHB What are the best definitions for limsup and liminf in sequences?

[Poirot1]

I am trying to learn about limsup and liminf for sequences in R. Walter rudin in his book defines them to be the sup and inf of the set of subsequential limits (possibly including + and - infinity). Wikipedia defines limsup to be the smallest number such that any greater number is an eventual upper bound of the sequence.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'.

 

[chisigma]

I am trying to learn about limsup and liminf for sequences in R. Walter rudin in his book defines them to be the sup and inf of the set of subsequential limits (possibly including + and - infinity). Wikipedia defines limsup to be the smallest number such that any greater number is an eventual upper bound of the sequence.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'.

In my opinion Wikipedia definition is clear and not ambigous so that I suggest to adopt it... if the sequence is umbounded on the upper part then the upper limit simply doesn't exist and if the sequence is umbounded on the lower part then the lower limit simply doesn't exist...

Kind regards

$\chi$ $\sigma$