Inequality of Lp-norm with supremum

Question

I'm reading a paper about the universal approximation of radial basis function (RBF) networks [1].

I cannot obtain the first inequality of Lemma 1 (right after "with $q$ defined by ..."), as shown in the below picture. Is there anyone who can derive this inequality?

Thanks.

[1] J. Park and I. W. Sandberg, “Universal Approximation Using Radial-Basis-Function Networks,” Neural Computation, vol. 3, no. 2, pp. 246–257, Jun. 1991, doi: 10.1162/neco.1991.3.2.246.