Proof of Bounded Sequences in a Separable Real Hilbert Space Have Weakly Convergent Subsequences

We use The Cauchy-Schwarz Inequality in a Real Inner Product Space, the identities of Elementary Identities in a Real Inner Product Space, and for real sequences Arithmetic of Limits of Real Sequences, Order Properties of Limits of Real Sequences and Every Cauchy Sequence of Real Numbers Converges. First, $0\le|x_{1}|\le C$, so $0\le C$.

A dense sequence. Let $D\subseteq H$ be countable and dense (Separable Metric Space). $D$ is nonempty, since $H\ne\varnothing$ and the closure of $\varnothing$ is $\varnothing$ (Closure of a Subset of a Topological Space: no point has every open neighbourhood meeting $\varnothing$, $H$ itself being open). By Countable Set there is a sequence $(y_{k})_{k\in\mathbb{N}}$ whose set of terms is $D$.

Diagonal extraction. Put $a_{m,k}=\langle x_{m},y_{k}\rangle$. By Cauchy-Schwarz and claim 5 of Elementary Arithmetic in an Ordered Field, $|a_{m,k}|\le|x_{m}|\,|y_{k}|\le C|y_{k}|$ for all $m,k$, so The Diagonal Subsequence Lemma for Bounded Real Arrays provides a strictly increasing $(n_{j})_{j\in\mathbb{N}}$ such that $(\langle x_{n_{j}},y_{k}\rangle)_{j}$ converges for every $k$. Write $z_{j}=x_{n_{j}}$; then $|z_{j}|\le C$ for all $j$.

Convergence against every vector. Let $y\in H$. We show that $(\langle z_{j},y\rangle)_{j}$ is a Cauchy sequence of real numbers. Let $\varepsilon>0$. Since $y$ lies in the closure of $D$, Sequential Characterization of the Closure in a Metric Space gives a sequence in $D$ converging to $y$, hence some $k$ with $|y-y_{k}|<\varepsilon/(3(C+1))$. The convergent sequence $(\langle z_{j},y_{k}\rangle)_{j}$, with limit $L$ say, satisfies: there is $J$ with $|\langle z_{j},y_{k}\rangle-L|<\varepsilon/6$ for $j\ge J$, hence $|\langle z_{j},y_{k}\rangle-\langle z_{i},y_{k}\rangle|\le|\langle z_{j},y_{k}\rangle-L|+|L-\langle z_{i},y_{k}\rangle|<\varepsilon/3$ for $i,j\ge J$ (claim 5 of Properties of the Absolute Value in an Ordered Field). For $i,j\ge J$, by bilinearity, the triangle inequality and Cauchy-Schwarz,

using $C/(C+1)\le 1$ (claim 5 of Elementary Arithmetic in an Ordered Field applied to $C\le C+1$ with the nonnegative multiplier $(C+1)^{-1}$). Hence the sequence is Cauchy and converges by Every Cauchy Sequence of Real Numbers Converges (the two notions of Cauchy sequence and of convergence for real sequences being those of Cauchy Sequence of Real Numbers and Limit of a Sequence of Real Numbers, read as in Convergence and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space). Let $\ell(y)$ denote its limit, unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences.

The limit functional. For $y,y'\in H$ and $\lambda\in\mathbb{R}$, $\langle z_{j},y+y'\rangle=\langle z_{j},y\rangle+\langle z_{j},y'\rangle$ and $\langle z_{j},\lambda y\rangle=\lambda\langle z_{j},y\rangle$ (Elementary Identities in a Real Inner Product Space §bilinear), so claims 1 and 3 of Arithmetic of Limits of Real Sequences and uniqueness of limits give $\ell(y+y')=\ell(y)+\ell(y')$ and $\ell(\lambda y)=\lambda\ell(y)$: $\ell$ is a linear functional. Moreover $|\langle z_{j},y\rangle|\le C|y|$ for all $j$, so $|\ell(y)|\le C|y|$ by claim 4 of Order Properties of Limits of Real Sequences and claim 1 there (against the constant sequence $(C|y|)$); thus $\ell$ is bounded. By The Riesz Representation Theorem for a Real Hilbert Space §existence there is $x\in H$ with $\ell(y)=\langle y,x\rangle=\langle x,y\rangle$ for every $y\in H$.

Conclusion. For every $y\in H$ the sequence $(\langle z_{j},y\rangle)_{j}$ converges to $\ell(y)=\langle x,y\rangle$; that is, $(x_{n_{j}})_{j}$ converges weakly to $x$. Finally $|x|\le C$ by Elementary Properties of Weak Convergence in a Real Inner Product Space §norm-bound.