The Scaling Set of an Open Convex Neighbourhood of the Origin Has an Infimum

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $X$ be a real normed space with zero vector $0_{X}$, let $U\subseteq X$ be open and convex with $0_{X}\in U$, and for $x\in X$ let

$$T_{x}=\{t\in\mathbb{R}:t>0\text{ and }t^{-1}x\in U\}.$$

(i) Large scalings. There is a real $r>0$ with $B(0_{X},r)\subseteq U$. For every such $r$ and every $x\in X$, every real $t>r^{-1}\lVert x\rVert$ lies in $T_{x}$.

(ii) Closed upwards. For $x\in X$, if $s\in T_{x}$ and $t>s$, then $t\in T_{x}$.

(iii) Existence of the infimum. For every $x\in X$, $T_{x}$ is nonempty and bounded below, with $0$ as a lower bound; hence it has an infimum, and $\inf T_{x}\ge0$.