Eliashberg–Gromov rigidity theorem ($\mathrm{Symp}(M,\omega )\text{ is }{C}^0\text{-closed in }\mathrm{Diff}(M)$)

A locally uniform limit of symplectomorphisms that is a smooth diffeomorphism is symplectic. By capacity preservation under uniform limits and linear capacity rigidity, its derivative is symplectic or anti-symplectic. Repeat this argument after taking the product of every map with the identity on a symplectic plane. An anti-symplectic derivative on the original factor and a symplectic derivative on the added factor would give neither sign on the product, a contradiction. The argument uses symplectic capacities in the original and stabilized dimensions.