Sequentially Strict Maxima and Minima on a Subset of a Metric Space

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space, let $A\subseteq X$, let $u:A\to\mathbb{R}$, and let $\bar{x}\in A$. A sequence in $X$ is said to converge when it converges in $(X,d)$.

1. (Sequentially strict maximum)(i) The function $u$ attains a sequentially strict maximum on $A$ at $\bar{x}$ if

$$u(x)\le u(\bar{x})\qquad\text{for every }x\in A,$$

and every sequence $(x_{m})_{m\in\mathbb{N}}$ in $A$ for which the sequence $(u(x_{m}))_{m\in\mathbb{N}}$ of real numbers converges to $u(\bar{x})$ converges to $\bar{x}$.

2. (Sequentially strict minimum)(ii) The function $u$ attains a sequentially strict minimum on $A$ at $\bar{x}$ if

$$u(\bar{x})\le u(x)\qquad\text{for every }x\in A,$$

and every sequence $(x_{m})_{m\in\mathbb{N}}$ in $A$ for which the sequence $(u(x_{m}))_{m\in\mathbb{N}}$ of real numbers converges to $u(\bar{x})$ converges to $\bar{x}$.