Exhausting Sequence of Finite-Dimensional Subspaces of a Real Hilbert Space

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, let $\mathbb{N}$ be the set of natural numbers, and let $H$ be a real Hilbert space with norm $|\cdot|$.

A sequence $(H_{n})_{n\in\mathbb{N}}$ of closed linear subspaces of $H$, each of which is a vector space over $\mathbb{R}$ under the restricted operations of $H$ by claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product, is an exhausting sequence for $H$ if(i)

1.(ii) $H_{n}\subseteq H_{n+1}$ for every $n\in\mathbb{N}$;

2.(iii) each $H_{n}$ is finite-dimensional as a vector space over $\mathbb{R}$;

3.(iv) for every $x\in H$ and every real $\varepsilon>0$ there exist $n\in\mathbb{N}$ and $y\in H_{n}$ with $|x-y|<\varepsilon$.