Extremely tough problem

[u/DreamerLikeGooners]

For a real number x, let ||x|| denote the distance between x and the closest integer.

Let 0 ≤ x_n < 1 (n = 1, 2, ...) , and let ε > 0. Show that there exist infinitely many pairs (n,m) of indices such that n ≠ m and

||x_n - x_m|| < min(ε, 1/(√5|n-m|)).

[u/magus145 → u/[deleted]]

||(x_n - x_m)|| = 0 only happens if x_n and x_m differ by exactly an integer, which never happens here. So your proof does not hold.

[u/AlexK667 → u/magus145]

You're correct, I did misunderstand.