Interchanging integral and sum

I have the following dilemma - I wish to prove rigourously the following:$$\int_{0}^{1} \int_{0}^{1} \frac{1}{1-xy} dxdy = \sum_{n\geq1} \frac{1}{n^2} $$Here the obvious strategy is to use the power series:$$\int_{0}^{1} \int_{0}^{1} \sum_{n\geq0} (xy)^n dx dy$$And the natural step would be to interchange the sum twice, first yielding us:$$\int_{0}^{1} \sum_{n\geq0} \int_{0}^{1} (xy)^n dx dy$$$$=\int_{0}^{1} \sum_{n\geq0} \int_{0}^{1} (xy)^n dx dy$$$$=\int_{0}^{1} \sum_{n\geq0} \frac{y^n}{n+1} dy$$Repeating this step gives us the wanted result. How can I justify interchanging the two steps without something like the Lebesgue dominated convergence theorem. Showing uniform convergence would be ideal. Thanks!