Exhausting Sequences of Finite-Dimensional Subspaces in a Separable Real Hilbert Space, and Their Projections

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 inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$ and zero vector $0_{H}$. Exhausting sequences for $H$ are as defined there. Then the following hold.

1. (Existence)(i) If $(H,d)$ is separable, then there exists an exhausting sequence $(H_{n})_{n\in\mathbb{N}}$ for $H$.

For the remaining claims let $(H_{n})_{n\in\mathbb{N}}$ be an exhausting sequence for $H$, let $P_{n}=P_{H_{n}}$ be the orthogonal projection of Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space, and let $Q_{n}:H\to H$ be given by $Q_{n}x=x-P_{n}x$.

2. (Projections and tails)(ii) For every $n\in\mathbb{N}$ and $x\in H$: $P_{n}$ and $Q_{n}$ are linear, $|P_{n}x|\le|x|$, $|Q_{n}x|\le|x|$, $Q_{n}x$ belongs to the orthogonal complement of $H_{n}$, $\langle P_{n}x,Q_{n}x\rangle=0$, and $|Q_{n+1}x|\le|Q_{n}x|$.

3. (Pointwise convergence)(iii) For every $x\in H$ the sequence $(P_{n}x)_{n\in\mathbb{N}}$ converges to $x$ and $(Q_{n}x)_{n\in\mathbb{N}}$ converges to $0_{H}$ in $(H,d)$.

4. (Along convergent sequences)(iv) If $(x_{n})_{n\in\mathbb{N}}$ is a sequence in $H$ converging to $x\in H$, then $(P_{n}x_{n})_{n\in\mathbb{N}}$ converges to $x$ and $(Q_{n}x_{n})_{n\in\mathbb{N}}$ converges to $0_{H}$.