For a recurrence $a_n = c_1 a_{n-1} + c_2 a_{n-2} + \dots + c_k a_{n-k}$, if the roots of the characteristic equation are $r_1, r_2, \dots, r_k$ where all $|r_i| < 1$, what can be inferred about the limit $\lim_{n \to \infty} a_n$?