If is unbounded above, take and find recursively: Given there exists such that for all , because otherwise taking would be a bound of .

If then is unbounded, and by the previous exercise has an increasing or decrasing subsequence.

If is irrational, an integer and were a rational number, since is a field then would be a rational number but . therefore is irrational.

The answer to the limit is five, although I use the fact that you want the limit of the -norm in Euclidean 3-space of the vector which is the -norm of which is . If you've seen this I'll elaborate, if not then use this only as a guide to the answer.

And finally, No, there cannot exist Cauchy sequences which do not converge because is complete