How to show there is no point in boundary for a convergent net in a closure?

Question

Let $X$ be a topological space. Let $M\subseteq X$. Let $(x_i)_{i\in I}$ be a net in $\overline{M}$ such that $x_i\rightarrow x\in\overline{M}$ and $x_i\neq x$ for all $i$. Then how to show $x_i\in M$ for all $i$?

Thanks in advance!

Answer 1

This is false. Let $X=\mathbb R^{2}, M$, the open unit disk in $X$ [i.e. $M=\{(a,b):a^{2}+b^{2}<1\}$] and $x_n=(\cos (\frac 1 n),\sin (\frac 1 n))$. Then $x_n \to x=(1,0)$, $x_n \neq x$ for all $n$ and no $x_n \in M$.