On sufficient conditions for existence of norm convergent subsequences in dual Banach spaces

Question

Let $\mathcal{B}$ be a Banach space with dual $\mathcal{B}^{*}$.

Let $\{a_{n}\}$ be a norm bounded sequence in $\mathcal{B}^{*}$

such that $||a_{n}- a_{n+1}||_{\mathcal{B}^{*}}\rightarrow 0.$

My questions are:

1. Is it true that $\{a_{n}\}$ has a subsequence converging in the norm of $\mathcal{B}^{*}$?

2. Assuming 1) is not true, are there any known sufficient conditions that would guarantee the existence of a norm convergent subsequence of $\{a_{n}\}$?