Exponeitiation by `$\dagger$' indicates the Moore-Penrose pseudo-inverse and the norms are the usual matrix norm induced by the Euclidean norm. Let $rank(A)\leq rank(B)$. I'm wondering whether one can upper bound $\|A(A^{\dagger}-B^{\dagger})\|$ in terms of $\|A^{\dagger}\|$ and $\|A-B\|$?
I know that this is possible when $rank(A)= rank(B)$ and that it is not possible when $rank(B)<rank(A)$ and $A$ has full column rank.