  The method of alternating projections involves orthogonally projecting an
element of a Hilbert space onto a collection of closed subspaces. It is known
that the resulting sequence always converges in norm if the projections are
taken periodically, or even quasiperiodically. We present proofs of such well
known results, and offer an original proof for the case of two closed
subspaces, known as von Neumann's theorem. Additionally, it is known that this
sequence always converges with respect to the weak topology, regardless of the
order projections are taken in. By focusing on projections directly, rather
than the more general case of contractions considered previously in the
literature, we are able to give a simpler proof of this result. We end by
presenting a technical construction taken from a recent paper, of a sequence
for which we do not have convergence in norm.
