Let $K$ be a bounded, closed, and convex subset of a Banach
space. For a Lipschitzian self-mapping $A$ of $K$ , we denote by
$\mathrm{Lip}(A)$ its Lipschitz constant. In this paper, we establish a
convergence property of infinite products of Lipschitzian
self-mappings of $K$ . We consider the set of all sequences
$\{A_t\}_{t=1}^{\infty}$ of such self-mappings with the property
$\mathrm{limsup}_{t \rightarrow \infty} \mathrm{Lip}(A_t) \le 1$ . Endowing it
with an appropriate topology, we establish a weak ergodic
theorem for the infinite products corresponding to generic
sequences in this space.