Let $E$ a Banach Space. Let Y:=\{h:E\to\mathbb{R} \ : \ h \text{ bounded, Fréchet differentiable and Lipschitz} \} . Let \|h\|_Y:=\|h\|_{\infty}+\|h'\|_{\infty}. Show that $(Y,\|\cdot\|_Y)$ is a Banach Space.
Edit: Ah, thanks for the tip, I have proved that is a norm only :S and I need the completeness.