The aim of this paper is to prove a counterpart of the Banach fixed point principle for mappings f : L∞ (X) → X, where X is a metric space and L∞ (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 → 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.