Sequential Banach-Alaoglu Theorem: Bounded Sequences in the Dual of a Separable Real Normed Space Have Weak-Star Convergent Subsequences

In the setting of The Real Numbers: Standing Notation and Background, let $E$ with norm $\lVert\cdot\rVert$ be a real normed space whose metric space $(E,d)$ of Real Normed Space and Real Banach Space §distance is separable, let $E^{*}$ be its dual space with the dual norm $\lVert\cdot\rVert_{E^{*}}$, let $M\in\mathbb{R}$ be nonnegative, and let $(\ell_{m})_{m\in\mathbb{N}}$ be a sequence in $E^{*}$ with $\lVert\ell_{m}\rVert_{E^{*}}\le M$ for every $m\in\mathbb{N}$.

(Weak-star compactness)¶ There are a strictly increasing sequence $(m_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ and an $\ell\in E^{*}$ with $\lVert\ell\rVert_{E^{*}}\le M$ such that $(\ell_{m_{j}})_{j\in\mathbb{N}}$ converges weak-star to $\ell$.