In a Diagonal Hilbert Triple whose Weights Tend to Infinity, Bounded Sequences of the Form Space Have Subsequences Converging in the Ambient Space

In the settings of Real Hilbert Spaces: Standing Notation and Background and Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus, let $(H,V,A)$ be the diagonal Hilbert triple determined by an orthonormal basis $(e_{k})_{k\in\mathbb{N}}$ of a real Hilbert space $H$ and by weights $(\lambda_{k})_{k\in\mathbb{N}}$, with the norms $|\cdot|_{H}$ and $|\cdot|_{V}$ and the distance $d_{H}$ of $H$. Assume that the weights tend to infinity: for every real number $M$ there is $k_{0}\in\mathbb{N}$ such that $M<\lambda_{k}$ for every $k\in\mathbb{N}$ with $k\ge k_{0}$.

¶Let $R\in\mathbb{R}$ be nonnegative and let $(x_{j})_{j\in\mathbb{N}}$ be a sequence in $V$ with $|x_{j}|_{V}\le R$ for every $j\in\mathbb{N}$. Then there are natural numbers $j_{1}<j_{2}<j_{3}<\cdots$ and a point $x\in V$ with $|x|_{V}\le R$ such that $(x_{j_{i}})_{i\in\mathbb{N}}$ converges to $x$ in the metric space $(H,d_{H})$.