One Modulus and a Sum Bound for a Finite Family

We work in the setting of Euclidean Space and Real Matrices: Standing Notation, whose notation is in force in every dimension; of it we use only the real numbers with their order and the associated strict order, the natural numbers $\mathbb{N}$ with successor map $S$, the initial segments $[n]$, and Euclidean space $\mathbb{R}^{n}$, of which only its elements as $n$-tuples of real numbers are used, and no matrix notation at all.

Let $(X,d)$ and $(Y,d_{Y})$ be metric spaces, let $A\subseteq X$ and let $x_{0}\in A$. For a natural number $n$ and an $n$-tuple $a\in\mathbb{R}^{n}$ with components $a_{k}$ we write $\sum_{k=1}^{n}a_{k}$ for the finite sum of its components, and we put

$$\sigma_{n}=\sum_{k=1}^{n}1 .$$

Then the following hold.

1. (One modulus for a finite family)(i) Let $n$ be a natural number, let $f$ assign to each $k\in[n]$ a function $f_{k}:A\to Y$ that is continuous at $x_{0}$ relative to $A$, and let $\varepsilon\in\mathbb{R}$ be positive. Then there is a positive $\delta\in\mathbb{R}$ such that every $y\in A$ with $d(x_{0},y)<\delta$ satisfies

$$d_{Y}\bigl(f_{k}(y),f_{k}(x_{0})\bigr)<\varepsilon\qquad\text{for every }k\in[n].$$

2. (The number of summands)(ii) For every natural number $n$ one has $1\le\sigma_{n}$, and $\sum_{k=1}^{n}c=\sigma_{n}c$ for every $c\in\mathbb{R}$.

3. (Uniform bound for a finite sum)(iii) Let $n$ be a natural number, let $a\in\mathbb{R}^{n}$ and let $t\in\mathbb{R}$ satisfy $a_{k}\le t$ for every $k\in[n]$. Then

$$\sum_{k=1}^{n}a_{k}\le\sigma_{n}t .$$