  We consider sequences $(B_k)_{k=0}^\infty$ of points obtained by projecting
back and forth between two manifolds $\M_1$ and $\M_2$, and give conditions
guaranteeing that the sequence converge to a limit $B_\infty\in\M_1\cap\M_2$.
Our motivation is the study of algorithms based on finding the limit of such
sequences, which have proven useful in a number of areas. The intersection is
typically a set with desirable properties, but for which there is no efficient
method of finding the closest point $B_{opt}$ in $\M_1\cap\M_2$. We prove not
only that the sequence of alternating projections converges, but that the limit
point is fairly close to $B_{opt}$, in a manner relative to the distance
$\|B_0-B_{opt}\|$, thereby significantly improving earlier results in the
field. A concrete example with applications to frequency estimation of signals
is also presented.
