# Convergence of a sequences of projecting point on a set of hyperplanes

How to prove next statement? Consider finite number intersecting hyperplanes in N dimensional Euclidean space and some starting point $x_0$. Point $x_i$ is obtained from point $x_{i-1}$ by projecting to some hyperplane, so $x_i = P_k(x_{i-1})$ where $P_k(\cdot)$ is orthogonal projection on hyperplane k. Given that we choose hyperplanes in some order, the sequence of point $x_0, x_1, \dots$ will converges. Hyperplanes go though zero and total number of them is N-1, so intersection of all of them is a line.

Also,I would really appreciate if somebody can give reference to similar statement in literature.

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Are you supposing that the number of hyperplanes is finite? – user40276 Jan 22 '13 at 21:59
Yes, the number of hyperplanes is finite. – ashim Jan 22 '13 at 22:01
You presumably mean hyperplanes through the origin, not affine hyperplanes, because if you had two parallel hyperplanes then you could project a point back and forth between them and the sequence wouldn't converge. Also "projecting operator" presumably means orthogonal projection. – Andreas Blass Jan 22 '13 at 23:06
Yes, you are right. – ashim Jan 22 '13 at 23:08