Sequential Form of the Continuity of a Second-Order Equation Operator

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation on which it rests, is in force in a dimension $n$, a natural number with $1\le n$.

Let $U\subseteq\mathbb{R}^{n}$ be open, let $F$ be a second-order equation operator on $U$, let $x_{0}\in U$, let $r_{0}\in\mathbb{R}$, let $p_{0}\in\mathbb{R}^{n}$ and let $X_{0}\in\mathcal{S}(n)$, and suppose that $F$ is continuous at $(x_{0},r_{0},p_{0},X_{0})$. Convergence of a sequence is understood in $(\mathbb{R}^{n},d_{E})$ for sequences in $\mathbb{R}^{n}$, in $\bigl(\mathcal{S}(n),d_{\mathcal{S}(n)}\bigr)$ for sequences in $\mathcal{S}(n)$, and in $(\mathbb{R},d_{\mathbb{R}})$ for sequences of real numbers, where $d_{\mathbb{R}}$ is the metric of The Absolute Value Metric on the Real Numbers.

Let $(y_{k})_{k\in\mathbb{N}}$ be a sequence in $U$ converging to $x_{0}$, let $(s_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}$ converging to $r_{0}$, let $(q_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{n}$ converging to $p_{0}$, and let $(Y_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{S}(n)$ converging to $X_{0}$.

Then the sequence of real numbers $\bigl(F(y_{k},s_{k},q_{k},Y_{k})\bigr)_{k\in\mathbb{N}}$ converges to $F(x_{0},r_{0},p_{0},X_{0})$.