[math] \newcommand{\mathds}{\mathbb}[/math]

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Given two projections [math]P,Q\in B(H)[/math], with [math]H[/math] being infinite dimensional, find an elementary proof for the fact that we have, for any [math]x\in H[/math],

[[math]] (PQ)^nx\to (P\wedge Q)x [[/math]]

but the operators [math](PQ)^n[/math] do not necessarily converge in norm.