Problem: Let $a_1, a_2, \cdots$ be an infinite sequence of positive reals such that $$ |a_i - a_j| > \frac{1}{i+j}, \quad \forall i \ne j. $$ Show that $\sup_i a_i \ge 1$.
I am able to do this problem if the sequence is increasing...because then I get something like,
$$a_i > \frac{1}{3}+ \frac{1}{5} + \cdots \frac{1}{2i-1}.$$
However, when the sequence might oscillate I don't know how to show that a subsequence goes close to something at least $1$.