TITLE: Sum of Singular Values of (A+B)
QUESTION [11 upvotes]: How we can prove that:
$$\sum_{i=1}^n\sigma_i(A+B)\leq\sum_{i=1}^n\sigma_i(A)+\sum_{i=1}^n\sigma_i(B)$$
Where $\sigma_i$s are singular values  $\sigma_1\geq\sigma_2\geq\cdots\geq\sigma_n\geq0$ .

REPLY [7 votes]: As we know the sum of singular values is equivalent to unitarily invariant norms. Matrix Analysis Book So instead of proving the $$\sum_{i=1}^n\sigma_i(A+B)\leq\sum_{i=1}^n\sigma_i(A)+\sum_{i=1}^n\sigma_i(B)$$
you can show $\||A+B|\|\leq \||A|\|+\||B|\|$ and the latter one is true because of norm property.