I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$:


$\psi$ maps continuously from $\mathbb{R}_k$ to a real number. $G_j$ are envelopes for the function classes in which $g_j$ live, though I don't think this is important.

The expression looks a bit like the opposite of the triangle inequality for L2 norms.

The expression clearly holds for k=1, but an attempt at an induction didn't get me anywhere.

EDIT: observe also that the The CR inequality gets us nowhere, only making LHS bigger:



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