Krdytkyu

$textbf{Problem}$ Prove that $W_0^{1,p}(Omega)$ is a Banach space where $Omega$ be an open and bounded set in $mathbb{R}^n$

$textbf{Proof}$ $quad $Let ${u_n}$ be the Cauchy Sequence in $W_0^{1,p}(Omega)$. Then, ${u_n}$ be also the Cacuhy Sequence in $W^{1,p}(Omega)$. Since $W^{1,p}(Omega)$ is a Banach space, there exists $u in W^{1,p}(Omega)$ such that $Vert u-u_n Vert _{W^{1,p}(Omega)} rightarrow 0 $ as $n rightarrow infty$. We suffices to show that $u in W_0^{1,p}(Omega)$.

Since $u_n in W_0^{1,p}(Omega)$, there exists $phi_{n_j} in C^{infty}_{0}(Omega)$ such that $Vert u_n - phi_{n_j}Vert _{W^{1,p}(Omega)} rightarrow 0 $ as $n_j rightarrow 0$.

Thus,
begin{align*}
Vert u - phi_{n_j} Vert_{W^{1,p}(Omega)}leq Vert u-u_k Vert_{W^{1,p}(Omega)}+Vert u_k-u_nVert_{W^{1,p}(Omega)}+Vert u_n-phi_{n_j}Vert_{W^{1,p}(Omega)}
end{align*}
(i) There exists $N_1>0$ such that
begin{align*}
Vert u-u_kVert_{W^{1,p}(Omega)}<epsilon/3
end{align*}
for $k>N_1$. ($u_n$ converge to $u$ in $W^{1,p}(Omega)$)

(ii) There exists $N_2>0$ such that
begin{align*}
Vert u_k-u_n Vert_{W^{1,p}(Omega)}<epsilon/3
end{align*}
for $n,k>N_2$. ($u_n$ Cauchy sequence in $W^{1,p}(Omega)$)

(iii) There exists $N_3>0$ such that
begin{align*}
Vert u_n-phi_{n_j}Vert _{W^{1,p}(Omega)}<epsilon/3
end{align*}
for $n_j>N_3$. ($u_n in W^{1,p}_0(Omega)$)

Consequently, $phi_{n_j} in C^{infty}_0(Omega)$ converge to $u$ in $W^{1,p}(Omega)$. i.e, $uin W^{1,p}_0(Omega)$.

I'm not sure my proof is right....

I want to know where my proof is wrong..

Any help is appreciated....

Thank you!

I think your proof is right.
But according to the definition of $W^{1,p}_0{Omega}$, which is $W^{1,p}_0{Omega}$ is the completion of $C_c^{infty}(Omega)$ in $W^{1,p}(Omega)$. So I think it's no need to proof $W^{1,p}_0(Omega)$ is a Banach space, because it's natural.