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

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 8.3 of Billingsley to the asymptotic tightness in $\ell^{+\infty}([a,b])$, the class of real bounded functions defined on $[a,b]$ with $a<b$, in the lines of Theorems 1.5.6 and 1.5.7 in van der vaart and Wellner.