Find the limit of sequence with respect to a given topology 
Find the limit of sequence $x_{n} = (\frac{1}{n},\frac{1}{n})$ with respect to the topology $$\tau = \{(n,\infty)\times (n, \infty): n \in \mathbb{N} \} \cup \{\emptyset,\mathbb{R^2}\}.$$

My work. Let $x$ be a limit. 
If $x<0$, every number can be a limit, because in this case neighbourhood of $x$ is $\mathbb{R}$.
If $x > 0$. For example, if $x = (0,0)$, neighbourhood looks like $(0,\infty)\times (0, \infty)$. So, here is not limit for sequence $x_n$?? Or limit equals $(1,1)$??

The only neighbourhood of $x_n$ for $n>1$ is $\mathbb{R}^2$. Since for any point in $\mathbb{R}^2\setminus((1,\infty)\times(1,\infty))$, $\mathbb{R}^2$ is the only neighbourhood incidentally containing infinitely many points of $\{x_n\}$.  points $\mathbb{R}^2\setminus((1,\infty)\times(1,\infty))$ is set of limit points of $\{x_n\}$.
