The Banach-Alaoglu says that the closed unit ball in a dual space $X^*$ of a normed space $X$ is compact in the weak-star topology.
The sequential Banach-Alaoglu says that the closed unit ball in a dual space of a separable normed space is sequentially compact in the weak-star topology.
According to page 71, example 5.3.1 ii) here, the assumption of separability cannot be dropped in the sequential Banach-Alaoglu. The proof makes sense to me. However I also reason that the (strong) Banach-Alaoglu covers the sequential Banach-Alaoglu as a special case. So by Banach Alaoglu, shouldn't the sequential Banach Alaoglu theorem still hold true for $X$ inseparable?