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

Each result cited is universally quantified over the data in its own statement. Elementary arithmetic and order facts in $\mathbb{R}$, the properties of $|\cdot|$, the limit laws, the order properties of limits and the uniqueness of limits are those put in force by The Real Numbers: Standing Notation and Background §background and The Real Numbers: Standing Notation and Background §sequences. By The Dual of a Real Normed Space is a Real Banach Space, and the Dual Norm is the Least Bound §bound, every $\ell\in E^{*}$ satisfies $|\ell(v)|\le\lVert\ell\rVert_{E^{*}}\lVert v\rVert$ for every $v\in E$; hence

as $0\le\lVert v\rVert$ by Real Normed Space and Real Banach Space §norm. For a linear functional $\ell$ and $u,w\in E$ one has $\ell(u-w)=\ell(u)-\ell(w)$ by additivity and homogeneity, since $u-w=u+(-1)w$ by claim 5 of Elementary Identities in a Vector Space.

Step 1 (A dense sequence). By Separable Metric Space there is a countable set $D\subseteq E$ that is dense in $(E,d)$, so every $v\in E$ lies in the closure of $D$, and by claim 3 of Characterization of the Closure in a Metric Space by Open Balls, for every positive $\eta$ there is $e\in D$ with $\lVert v-e\rVert=d(v,e)<\eta$ (Real Normed Space and Real Banach Space §distance). Applied to $v=0_{E}$ and $\eta=1$, this shows that $D$ is nonempty. By Countable Set there is a sequence $(e_{k})_{k\in\mathbb{N}}$ in $D$ whose set of terms is $D$.

Step 2 (Diagonal subsequence). Put $a_{m,k}=\ell_{m}(e_{k})$ for $m,k\in\mathbb{N}$. By ($*$), $|a_{m,k}|\le M\lVert e_{k}\rVert$ for every $m$, so The Diagonal Subsequence Lemma for Bounded Real Arrays, with $R_{k}=M\lVert e_{k}\rVert$, gives a strictly increasing sequence $(m_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ such that for every $k$ the sequence $(\ell_{m_{j}}(e_{k}))_{j\in\mathbb{N}}$ converges; let $L_{k}$ be its limit.

Step 3 (Convergence at every vector). Let $v\in E$; we show that $(\ell_{m_{j}}(v))_{j\in\mathbb{N}}$ is a Cauchy sequence. Let $\varepsilon$ be positive. First choose, by Step 1, an element of $D$ with distance less than $\varepsilon/(3(M+1))$ from $v$; it is $e_{k}$ for some $k$. Then choose $J\in\mathbb{N}$ with $|\ell_{m_{j}}(e_{k})-L_{k}|<\varepsilon/6$ for every $j\ge J$. For $i,j\ge J$, the triangle inequality gives $|\ell_{m_{i}}(e_{k})-\ell_{m_{j}}(e_{k})|<\varepsilon/3$, and by linearity and ($*$),

So the sequence is Cauchy, and it converges by Every Cauchy Sequence of Real Numbers Converges. Let $\ell(v)$ be its limit; this defines $\ell:E\to\mathbb{R}$.

Step 4 (The limit functional). For $u,v\in E$ and $\lambda\in\mathbb{R}$, $\ell_{m_{j}}(u+v)=\ell_{m_{j}}(u)+\ell_{m_{j}}(v)$ and $\ell_{m_{j}}(\lambda v)=\lambda\,\ell_{m_{j}}(v)$ for every $j$, so claims 1 and 3 of Arithmetic of Limits of Real Sequences and uniqueness of limits give $\ell(u+v)=\ell(u)+\ell(v)$ and $\ell(\lambda v)=\lambda\,\ell(v)$: $\ell$ is a linear functional. By ($*$), $-M\lVert v\rVert\le\ell_{m_{j}}(v)\le M\lVert v\rVert$ for every $j$, so claim 1 of Order Properties of Limits of Real Sequences, applied twice with constant sequences, gives $-M\lVert v\rVert\le\ell(v)\le M\lVert v\rVert$, that is, $|\ell(v)|\le M\lVert v\rVert$. Thus $M$ is a nonnegative bound for $\ell$, so $\ell\in E^{*}$ and $\lVert\ell\rVert_{E^{*}}\le M$ by The Dual of a Real Normed Space is a Real Banach Space, and the Dual Norm is the Least Bound §bound. Finally, $(\ell_{m_{j}})_{j\in\mathbb{N}}$ is a sequence in $E^{*}$ with $\ell_{m_{j}}(v)\to\ell(v)$ for every $v\in E$, which is weak-star convergence to $\ell$.