Let $(Z_n)_{n\ge 1}$ be Borel random elements of $\ell^\infty(F)$, or more generally let them be maps into $\ell^\infty(F)$ interpreted through the usual outer-probability convention and assumed asymptotically measurable. The sequence $(Z_n)$ is tight if for every $\varepsilon>0$ there exists a compact set $K\subset \ell^\infty(F)$ such that \begin{align*} \inf_{n\ge 1}\mathbb P(Z_n\in K)\ge 1-\varepsilon. \end{align*} It is asymptotically tight if for every $\varepsilon>0$ there exists a compact set $K\subset \ell^\infty(F)$ such that \begin{align*} \liminf_{n\to\infty}\mathbb P(Z_n\in K)\ge 1-\varepsilon. \end{align*} If $Z_n$ is asymptotically tight, asymptotically measurable in the non-Borel case, and its finite-dimensional distributions converge to those of a tight Borel random element $Z$ of $\ell^\infty(F)$, then $Z_n\xrightarrow{d}Z$ in $\ell^\infty(F)$.