Extension: Limits of Subsequences

Let $\{a_n\}$ be a sequence for $n \ge 1$ such that $\lim_{n \to \infty} a_n = 1$. Now let $\{b_n\}$ be a sequence which satisfies $b_{2n} = a_n$ for all $n \ge 1$. Which of the following might be true of $\{b_n\}$?