  We consider a dynamic programming problem with arbitrary state space and
bounded rewards. Is it possible to define in an unique way a limit value for
the problem, where the "patience" of the decision-maker tends to infinity ? We
consider, for each evaluation $\theta$ (a probability distribution over
positive integers) the value function $v_{\theta}$ of the problem where the
weight of any stage $t$ is given by $\theta_t$, and we investigate the uniform
convergence of a sequence $(v_{\theta^k})_k$ when the "impatience" of the
evaluations vanishes, in the sense that $\sum_{t} |\theta^k_{t}-\theta^k_{t+1}|
\rightarrow_{k \to \infty} 0$. We prove that this uniform convergence happens
if and only if the metric space ${v_{\theta^k}, k\geq 1}$ is totally bounded.
Moreover there exists a particular function $v^*$, independent of the
particular chosen sequence $({\theta^k})_k$, such that any limit point of such
sequence of value functions is precisely $v^*$. Consequently, while speaking of
uniform convergence of the value functions, $v^*$ may be considered as the
unique possible limit when the patience of the decision-maker tends to
infinity. The result applies in particular to discounted payoffs when the
discount factor vanishes, as well as to average payoffs where the number of
stages goes to infinity, and also to models with stochastic transitions. We
present tractable corollaries, and we discuss counterexamples and a conjecture.
