If ${a_n}$ is bounded and non-decreasing, prove that $liminf b_n = 0$, $b_n = n(a_{n+1} - a_{n})$
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Let ${a_n}$ be a bounded and non-decreasing sequence of reals, and $b_n = n(a_{n+1} - a_{n})$ . (a) Show that $liminf b_n = 0$ (b) Give an example of a sequence ${a_n}$ such that ${b_n}$ diverges. I know that $a_{n+1} geq a_{n}, forall n$ , Hence, $a_{n+1} - a_{n} geq 0, forall n$ . Also, we have that $exists B >0$ s.t $|a_n| leq B, forall n$ . Hence, $b_n geq 0$ and $|b_n| leq n2B$ If ${a_n}$ is bounded, then ${a_n}$ converges by Bolzano Weierstrass Theorem, since ${a_n} subseteq [-B,B]$ Now, since ${a_n}$ converges, there exists a subsequence ${a_{n_k}}$ of bounded variation, hence $forall epsilon >0, exists N >0, forall n_kgeq N$ , $|a_{n_{k+1}} - a_{n_k}| < epsilon$ But I can't proof that the $liminf b_n =0$ .