Are "boundary" conditions of a sequence of functions preserved in the limit?

Question

I have a bounded sequence of functions in $$A=\{\xi\in H^1_{loc}(\mathbb{R}),\ \ \lim_{x\rightarrow\infty}\xi(x)=1,\ \ \lim_{x\rightarrow-\infty}\xi(x)=0\},$$ say $\{\xi_n\}_{n\in\mathbb{N}}$. Since the functions are in $H^1_{loc}(\mathbb{R})$ they are continuous. By a diagonal argument I constucted a subsequence of $\{\xi_n\}_{n\in\mathbb{N}}$ which is convergent pointwise a.e. on $\mathbb{R}$ and weakly in $H^1(K)$ for any fixed compact $K$. Now this limit, say $\xi$, is in $H^1_{loc}(\mathbb{R})$ thus it is continuous. I have one question:
Does $\xi\in A$? I wanted to prove this by an $\epsilon-\delta$ argument but fixing $\epsilon>0$ gives me $M_n>0$ s.t. $\vert 1-\xi_n(x)\vert<\epsilon$, when $x>M_n$, but I have no additional informations on this sequence $M_n$.