# How to show that $\sigma_{k+1}$ is the $2$-norm (spectral) distance of $A$ to the nearest rank-$k$ matrix?

I want to show that $$\sigma_{k+1}$$ is the $$2$$-norm (spectral) distance of $$A$$ to the nearest rank-$$k$$ matrix, with $$\sigma_i$$ denote the $$i$$-th singular value of $$A$$.

Could someone please give me some hints? Much appreciated. Thanks.

• Thanks for your comments. I've edited my question. – Steve Dec 2 at 20:45