X1. Let an be a sequence in R, and let bk = sup{ak, ak+1, ak+2, ...}. Prove bk is a monotone decreasing sequence. (It's very intuitive that the statement is true, and proving it doesn't require anything "clever". This problem is just an exercise in giving a formal proof of something that is almost obvious.)
X2. Some of the notation in Proposition 2.36 is chosen poorly, in my opinion. Can you figure out what I have in mind? (I'm not referring to the subscript "j -> infinity" for lim inf and lim sup, which the author sometimes uses, and sometimes not.)