On Guttman's Proof That Squared Multiple Correlations are Lower Bounds for Communalities

Guttman's proof that squared multiple correlations are lower bounds for communalities is shown to be based on assumptions so unrealistic as to render the proof invalid, with sample correlation matrices, and, probably, with population correlation matrices. (While mathematically correct, the proof is based on assumptions which do not hold with real data.) But then, it does appear that the theorem almost always holds in practice.