 Fejér monotonicity

A sequence is Fejér monotone with respect to a nonempty set $C$ if $\Vert z^{k+1}-p\Vert \le \Vert z^k-p\Vert$ for every $p\in C$. It is bounded. If it has a cluster point $\bar{z}\in C$, its nonincreasing distance to $\bar{z}$ and that convergent subsequence force the entire sequence to converge to $\bar{z}$.