convergence of singular values

I jus want to know how to show that if a matrix X converge to Y ( with respect to any matrix norm) then the ith singular value of X converge to the ith singular value of Y.

Thank you

• Is this a homework problem? What have you tried so far? – MCT Mar 23 '14 at 0:12
• I'm reading an article about rank minimization where they use this fact, i think about caracteristic polynomial of transpose(X)*X, can we say that if a Polynom P converge to Q, then it's the same thing for the roots? – user47204 Mar 25 '14 at 9:40

This is stated by Weyl's inequality: for any two matrices $A$ and $\tilde A$ which are related by $A - \tilde A = E$, their corresponding singular values satisfy $$| \tilde\sigma_i - \sigma_i | \le \| E \|_2.$$