  Let $f:\mathbb{C}\rightarrow \mathbb{R}^3$ be complete Willmore immersion
with $\int_{\Sigma}|A_f|^2<+\infty$. We will show that if $f$ is the limit of
an embedded surface sequence, then $f$ is a plane. As an application, we prove
that if $\Sigma_k$ is a sequence of closed Willmore surface embedded in
$\mathbb{R}^3$ with $W(\Sigma_k)<C$, and if the conformal class of $\Sigma_k$
converges in the moduli space, then we can find a M\"obius transformation
$\sigma_k$, such that a subsequence of $\sigma_k(\Sigma_k)$ converges smoothly.
