I have a metric space $M$ and a convergent sequence $(x\_n)$ in $M$. I also have $a$ in $M$ such that:

1) the set $\{a, x\_1, x\_2, \dots, x\_n,\dotsc\}$ is closed.

2) the set $\{x\_1, x\_2, \dots, x\_n, \dots\}$ is neither open nor closed. 

What would this say about $\lim\_{n\to\infty} x\_n$? 
This is puzzling me.  

Would this limit exist? Any hints would be great!