How to decide convergence of double power series of the form $\sum\limits_{n \geq 0} \sum\limits_{m \geq 0} a_{mn} x_1^mx_2^n$ ? For example the power series $\sum\limits_n x_1^n x_2^n$ converges in the set $\{ (x_1,x_2): |x_1x_2|<1 \} \subset \mathbb{R}^2$.

In my efforts to improve my mathematical rigour I am trying to understand each precise step in the proofs I attempt as well as look at the solution for. In this instance I am trying to understand the process in establishing whether a point is a limit point in $\mathbb{R}$. As the title says I'm t...