reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then $$\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).$$ Here $A^\dagger$ denotes the pseudoinverse of a matrix.

I believe that this result is established by Golub and Zha in the course of their proof of Theorem 3.6 in "Perturbation Analysis of the Canonical Correlations of Matrix Pairs," but in a manner that's too messy to point to and say "here it is."

Unfortunately, this result also doesn't seem to follow readily from the results in Stewart's paper on perturbation theory for pseudoinverses and projections.

Is this result clearly established somewhere in the literature?

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