Proposition 7

Let E be a Hausdorff prehilbertian space and let $\Psi$ be a non-empty directed increasing set of non-empty, convex, complete subsets of E. Put $A = \bigcup_{H \in \Psi} H$ and suppose that the closure N of A is complete. Then N is convex and we have

It is clear that A is convex, hence its closure N is convex (II, p. 13). With the notations of prop. 6, $d(x, N) = \inf_{H \in \Psi} d(x, H)$, and consequently $d(x, N)$ is the limit of $d(x, H)$ with respect to the section filter of $\Psi$. Since $p_H(x) \in H$ and

it follows from cor. 1 of V, p. 11 that $p_H(x)$ tends to the projection $p_N(x)$ of $x$ onto N with respect to the section filter of $\Psi$.