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sup {x : a_n > x for infinitely many n}.

It is very hard to work with this definition. I'm used to the simpler one:

sup {a : there exists a subsequence of a_1, a_2, ... that converges to a}.

I'm trying to show that these two are equivalent. Denote by A and B the set in the first and second definition, respectively. For any subsequence that converges to a in B, there is a monotonic subsequence, say b_1, b_2, ... that converges to a. If b_1, b_2, ... is increasing, then every b_i is in A. If it is decreasing, then x = inf {b_1, b_2, ...} is in A. For every x in A, there is a corresponding convergent monotonic subsequence. This is all I can think of. Any tips?