Let be the set of all orthonormal subsets of the given Hilbert space , which is partially ordered by set inclusion. It is nonempty as it contains the empty set and thus by the maximal principle, it contains a maximal chain . Let be the union of . We shall show it is a maximal orthonormal subset. First, if are in , then either or . That is, any given two distinct elements in are contained in some in and so they are orthogonal to each other (and of course, is a subset of the unit sphere in ). Second, if for some in , then cannot be in and so is a chain strictly larger than , a contradiction.
Let be the set of all orthonormal subsets of the given Hilbert space , which is partially ordered by set inclusion. It is nonempty as it contains the empty set and thus by the maximal principle, it contains a maximal chain  Le.t be the union of . We shall show it is a maximal orthonormal subset. First, if are in , then either or . That is, any given two distinct elements in are contained in some in and so they are orthogonal to each other (and of course, is a subset of the unit sphere in ). Second, if for some in , then cannot be in and so is a chain strictly larger than , a contradiction.