Limits and Bounded Sequences of Symmetric Real Matrices

We work in the setting of Euclidean Space and Real Matrices: Standing Notation, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers, the absolute value $|\cdot|$, the initial segments $[n]$, sequences, and Euclidean space $\mathbb{R}^{n}$ with its sum and difference of points, dot product, Euclidean norm $\lVert\,\cdot\,\rVert$ and distance $d_{E}$, the real matrices, their differences and scalar multiples, the matrix-vector product and the identity matrix $I_{n}$, the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices, which contains $-aI_{n}$ for every real $a$, the positive semidefinite ordering $\preceq$, the norm $\lVert P\rVert$, the distance $d_{\mathcal{S}(n)}$ and the notions of convergence they determine, are all as fixed there.

Then the following hold.

1. (Bounded sequences have convergent subsequences) (i) Let $(X_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{S}(n)$ and let $R\in\mathbb{R}$ satisfy $\lVert X_{m}\rVert\le R$ for every $m\in\mathbb{N}$. Then there exist a strictly increasing map $\varphi:\mathbb{N}\to\mathbb{N}$ and $X\in\mathcal{S}(n)$ such that the sequence $\bigl(X_{\varphi(k)}\bigr)_{k\in\mathbb{N}}$ converges to $X$ in $\bigl(\mathcal{S}(n),d_{\mathcal{S}(n)}\bigr)$.

2. (Convergence of quadratic forms) (ii) Let $(X_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{S}(n)$ converging to $X\in\mathcal{S}(n)$ in $\bigl(\mathcal{S}(n),d_{\mathcal{S}(n)}\bigr)$. Then for every $z\in\mathbb{R}^{n}$ the sequence of real numbers $\bigl(z\cdot(X_{m}z)\bigr)_{m\in\mathbb{N}}$ converges to $z\cdot(Xz)$.

3. (The ordering passes to limits) (iii) Let $(X_{m})_{m\in\mathbb{N}}$ and $(Y_{m})_{m\in\mathbb{N}}$ be sequences in $\mathcal{S}(n)$ converging in $\bigl(\mathcal{S}(n),d_{\mathcal{S}(n)}\bigr)$ to $X$ and to $Y$ respectively, and suppose $X_{m}\preceq Y_{m}$ for every $m\in\mathbb{N}$. Then $X\preceq Y$.

4. (Two-sided order bounds give a norm bound) (iv) Let $X,C\in\mathcal{S}(n)$ and let $a\in\mathbb{R}$ satisfy $0\le a$. If $-aI_{n}\preceq X$ and $X\preceq C$, then $\lVert X\rVert\le a+\lVert C\rVert$.