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I want to prove the existence of QR decomposition for rank deficient matrices and I am stuck on the step which says that: for any rank-deficient matrix there is a sequence of full-column rank matrices $A_k$ such that $A_k \to A$ where $A$ is a full column rank matrix. I guess that the set of full column rank matrices is compact and therefore there exist a converging subsequence of matrices $A_k \to A$ but is it correct?

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