If $x_n\to y $ and $d(x_n, x)<\epsilon$, $\forall x_n$. Is it true $d(y, x)<\epsilon$? Let the sequence $\{x_n\}_{n=0}^\infty$ be given and for all $x_n$, we have $d(x_n, x)<\epsilon$ and $x_n\to y$.
Is it true that $d(y, x)<\epsilon$? ( It is important for me to know that  $d(y, x)\neq \epsilon$)

Try $x_n=1/n\in\Bbb R$, $y=0$ and $x=\epsilon$. (Start the sequence with $n$ sufficiently large.) ... So, no, you may well have $d(y,x) = \epsilon$.
