$(L_{c}(E,F),\left \| . \right \|_{E\rightarrow F})$ Banach $\Leftrightarrow (F,\left \| . \right \|_{F})$ Banach

Question

Let $(\varphi_{n}) \subset (L_{c}(E,F),\left \| . \right \|_{E\rightarrow F})$ Cauchy. $\left \| \varphi_{n}-\varphi_{p} \right \|_{E\rightarrow F}=sup_{\left \| x \right \|\leqslant 1}\left \| \varphi_n(x)-\varphi_{p}(x) \right \|_{F}\leq \epsilon$.
$(\varphi_{n})_{x \in B(0,1),n}$ converge continuously to $\varphi$ because bounded and F is banach.
$\forall x \in E$ but not in $B(0,1)$ $\varphi_{n}(x)=\varphi_{n}(\frac{x}{\left \| x \right \|})\times\left \| x \right \|\rightarrow \varphi(x)$. For $N\geqslant 0, \varphi_{N}$ is continuous, $\left \| \varphi(x)-\varphi(y) \right \| \leqslant \left \| \varphi(x)-\varphi_{N}(x) \right \|+\left \| \varphi_{N}(x)-\varphi_{N}(y) \right \|+\left \| \varphi_{N}(y)-\varphi(y) \right \|\leqslant 3\epsilon$, $\forall x,y \in E$ It is correct ?