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In this note, we extend a simple criteria for uniform tightness in $C(0,1)$,the class of real continuous functions defined on $(0,1)$, given in Theorem 8.3of Billingsley to the asymptotic tightness in $\ell^{+\infty}([a,b])$, theclass of real bounded functions defined on $[a,b] $ with $a<b$, in the lines ofTheorems 1.5.6 and 1.5.7 in van der vaart and Wellner.

A remark on the asymptotic tightness in $\ell^{\infty}([a,b])$