  The aim of this paper is to prove a counterpart of the Banach fixed point
principle for mappings $f: \ell_\infty(X) \to X$, where $X$ is a metric space
and $\ell_\infty(X)$ is the space of all bounded sequences of elements
from~$X$. Our result generalizes the theorem obtained by Miculescu and Mihail
in 2008, who proved a~counterpart of the Banach principle for mappings
$f:X^m\to X$, where $X^m$ is the Cartesian product of $m$ copies of $X$. We
also compare our result with a recent one due to Secelean, who obtained a
weaker assertion under less restrictive assumptions. We illustrate our result
with several examples and give an application.
