Proof of The Quadratic Form of a Real Square Matrix is Bounded on the Closed Unit Ball

Nonemptiness. Write $0_{\mathbb{R}^n}$ for the origin of $\mathbb{R}^n$. By claim 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\lVert0_{\mathbb{R}^n}\rVert=0$, and $0\le1$ by claim 1 of Elementary Arithmetic in an Ordered Field; hence $0_{\mathbb{R}^n}$ is one of the points admitted in the description of $Q_A$, and $Q_A$ is nonempty.

Upper bound. Let $\xi\in\mathbb{R}^n$ with $\lVert\xi\rVert\le1$. By claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ we have $|\xi_i|\le\lVert\xi\rVert$ for every $i\in[n]$, so $|\xi_i|\le1$ by transitivity of $\le$, which is a total order. By claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum,

Fix $i\in[n]$ and let $j\in[n]$. By claim 4 of Properties of the Absolute Value in an Ordered Field, $|A_{ij}\,\xi_i\,\xi_j|=|A_{ij}|\,|\xi_i|\,|\xi_j|$, and by claim 1 of that lemma each of the three factors is nonnegative. Applying claim 5 of Elementary Arithmetic in an Ordered Field to $|\xi_j|\le1$ with the nonnegative factor $|\xi_i|$ gives $|\xi_i|\,|\xi_j|\le|\xi_i|$, hence $|\xi_i|\,|\xi_j|\le1$ by transitivity; applying claim 5 of the same lemma again, now with the nonnegative factor $|A_{ij}|$, gives

Therefore, by claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers followed by claim 1 of that lemma,

Applying claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers to the outer sum, and then claim 1 of that lemma with the bound just obtained,

Since $\xi$ was an arbitrary point with $\lVert\xi\rVert\le1$, the real number $\sum_{i=1}^{n}\sum_{j=1}^{n}|A_{ij}|$ is an upper bound for $Q_A$. Being a nonempty subset of $\mathbb{R}$ with an upper bound, $Q_A$ has a least upper bound in $\mathbb{R}$ by the least upper bound property of the real numbers.