Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space, let $C\in\mathbb{R}$ be nonnegative, let $\omega$ be a modulus of continuity, and let $(u_{N})_{N\in\mathbb{N}}$ be a sequence of functions $u_{N}:X\to\mathbb{R}$ such that

$$|u_{N}(x)|\le C\quad\text{and}\quad|u_{N}(x)-u_{N}(y)|\le\omega\bigl(d(x,y)\bigr)\qquad\text{for all }N\in\mathbb{N}\text{ and }x,y\in X .$$

For each $x\in X$ the sequence $(u_{N}(x))_{N\in\mathbb{N}}$ is a bounded sequence of real numbers; let $\bar{u}(x)$ be its limit superior and $\underline{u}(x)$ its limit inferior. Then the following hold.

1. (Bounds)¶ For every $x\in X$, $|\bar{u}(x)|\le C$ and $|\underline{u}(x)|\le C$.

2. (Modulus)¶ For all $x,y\in X$,

$$|\bar{u}(x)-\bar{u}(y)|\le\omega\bigl(d(x,y)\bigr)\quad\text{and}\quad|\underline{u}(x)-\underline{u}(y)|\le\omega\bigl(d(x,y)\bigr).$$

3. (Extraction)¶ For every $x\in X$ there are natural numbers $N_{1}<N_{2}<N_{3}<\cdots$ such that $(u_{N_{j}}(x))_{j\in\mathbb{N}}$ converges to $\bar{u}(x)$, and natural numbers $N'_{1}<N'_{2}<N'_{3}<\cdots$ such that $(u_{N'_{j}}(x))_{j\in\mathbb{N}}$ converges to $\underline{u}(x)$.

4. (Uniform convergence on sequentially compact sets)¶ Let $K\subseteq X$ be sequentially compact, and suppose that $\bar{u}(x)=\underline{u}(x)$ for every $x\in K$. Then for every real $\varepsilon>0$ there is $N_{0}\in\mathbb{N}$ such that $|u_{N}(x)-\bar{u}(x)|\le\varepsilon$ for every $N\in\mathbb{N}$ with $N\ge N_{0}$ and every $x\in K$.