Let (Xn) be an unbounded sequence in R. If (Xn) is unbounded above, show that (Xn) has a subsequence (Xnk) with Xnk>k for all k in N.
Since it's unbounded from above, it has an unbounded increasing subsequence. Let X_n_1 be the first member of this subsequence greater than 1, X_n_2 be the first member of this subsequence after X_n_1 greater than 2, and so forth.