SCHWEITZER 1967 Problem 6

Let $A$ be a family of proper closed subspaces of the Hilbert space $H=l^2$ totally ordered with respect to inclusion (that is , if $L_1,L_2 \in A$ , then either $L_1\subset L_2$ or $L_2\subset L_1$ ). Prove that there exists a vector $x \in H$ not contaied in any of the subspaces $L$ belonging to $A$ .

B. Szokefalvi Nagy