Let $A$ and $B$ be two rank-$r$ real matrices of size $n×n$. Let $P_A$ and $P_B$ be the orthogonal projection operator onto the column spaces of $A$ and $B$ respectively. Show that $$||P_A-P_B|| \leq\frac{||A-B||}{\sigma_{min}(A)}$$ where $\sigma_{min}(A)$ is the minimum nonzero singular value of $A$, and $||\cdot||$ is the spectral norm.
So the left hand side is the distance between the range of matrix $A$ and $B$. However, I don't have an intuition of what right hand side is. I am quite confused about this relation. Can anyone give me some hints? Thanks in advance!