Elementary Properties of Sequentially Strict Extrema

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space, let $A\subseteq X$, let $u,v:A\to\mathbb{R}$, let $\bar{x}\in A$, let $c\in\mathbb{R}$ and let $\lambda\in\mathbb{R}$ with $0<\lambda$. Sequentially strict maxima and minima are as defined there. Write $-u$, $u+c$, $\lambda u$ and $u+v$ for the functions from $A$ to $\mathbb{R}$ whose values at $x\in A$ are $-u(x)$, $u(x)+c$, $\lambda\,u(x)$ and $u(x)+v(x)$. Then the following hold.

1. (Negation)(i) The function $u$ attains a sequentially strict maximum on $A$ at $\bar{x}$ if and only if $-u$ attains a sequentially strict minimum on $A$ at $\bar{x}$.

2. (Strictness and uniqueness)(ii) Suppose $u$ attains a sequentially strict maximum on $A$ at $\bar{x}$. Then $u(x)<u(\bar{x})$ for every $x\in A$ with $x\ne\bar{x}$. Consequently $\bar{x}$ is the only point of $A$ at which $u$ attains its maximum value, and in particular the only point of $A$ at which $u$ attains a sequentially strict maximum.

3. (Affine changes)(iii) If $u$ attains a sequentially strict maximum on $A$ at $\bar{x}$, then so do $u+c$ and $\lambda u$.

4. (Addition of a function maximised at the same point)(iv) Suppose $u$ attains a sequentially strict maximum on $A$ at $\bar{x}$ and that $v(x)\le v(\bar{x})$ for every $x\in A$. Then $u+v$ attains a sequentially strict maximum on $A$ at $\bar{x}$.

5. (Restriction)(v) Suppose $u$ attains a sequentially strict maximum on $A$ at $\bar{x}$, let $B\subseteq A$ with $\bar{x}\in B$, and let $u|_{B}$ denote the restriction of $u$ to $B$. Then $u|_{B}$ attains a sequentially strict maximum on $B$ at $\bar{x}$.