I want to show that $\|\cdot \|_{\rm op}$ (largest singular value) is a Lipschitz function on the set of $m \times n$ real matrices, $\mathbb{R}^{m \times n}$.
I see often people citing Lidskii's inequality to prove it, but I think the following also shows it: use the variational characterization of the operator norm: $$ \|A\|_{\rm op} = \sup_{\|x\| =\|y\| = 1} x^T Ay $$ Fix two matrices $A, B$, and without loss suppose $$ |\|A\|_{\rm op} - \|B\|_{\rm op}| = \|A\|_{\rm op} - \|B\|_{\rm op}. $$ Then, there is $\|u\| = \|v\| = 1$ so that $$ \|A\|_{\rm op} - \|B\|_{\rm op} \leq u^T(A - B) v \leq \|uv^T\|_F \|A-B\|_F = \|A-B\|_F. $$ Hence, $$ |\|A\|_{\rm op} - \|B\|_{\rm op}| \leq \|A-B\|_F. $$ Does this not imply that the operator norm is $1$-Lipschitz with respect to the usual (Frobenius) norm on $\mathbb{R}^{m \times n}$?