I am reading the paper "A. D. Ioffe, An invitation to tame optimization, SIAM J. Optim., 19 (2009), 1894–1917" and stuck at Proposition 3.1. The author claims (in the language of semi-algebraic geometry) that if $\varphi:\mathbb{R}^n\times\mathbb{R}^m\rightarrow\overline{\mathbb{R}}$ is a semi-algebraic function then the function $$\psi(x):=\sup_{\varepsilon>0}\inf_{0<\left\|h\right\|<\varepsilon}\varphi(x,h)$$ is also semi-algebraic. I tried to use the Tarski-Seidenberg Theorem to prove this claim but I could not. Although it is easy to show that $g(x)=\inf_{h\in S}\varphi(x,h)$ is semi-algebraic for any semi-algebraic set $S\subset \mathbb{R}^m$, it seems too hard for me to deal with $\inf_{0<\left\|h\right\|<\varepsilon}\varphi(x,h)$ where the infimum is taken respect to $\varepsilon$ which is not fixed.
I am very grateful to you if you could help me to overcome this obstacle. Thank you for your help!