  The notion of Fej\'er monotonicity is instrumental in unifying the
convergence proofs of many iterative methods, such as the Krasnoselskii-Mann
iteration, the proximal point method, the Douglas-Rachford splitting algorithm,
and many others. In this paper, we present directionally asymptotical results
of strongly convergent subsequences of Fej\'er monotone sequences. We also
provide examples to show that the sets of directionally asymptotic cluster
points can be large and that weak convergence is needed in infinite-dimensional
spaces.
