Limites de repr\'esentations cristallines

We provide an answer to two questions of Fontaine (in the unramified case). First, we show that a limit of crystalline representations, of bounded Hodge-Tate weights, is itself crystalline. Second, we show that every admissible filtered $\varphi$-module “comes from” a $(\varphi ,\mathrm{\Gamma })$-module of finite q-height.