Proving $(\sum_{i=0}^{\infty}x^{i})^2=\sum_{n=0}^{\infty} (n+1)x^n$

We are trying to evaluate:

$$\sum_{i=0,j=0}^{\infty}x^{i+j}=\sum_{n=0}^{\infty} (n+1)x^n$$

My attempt:

$$\sum_{i,j=0}^\infty x^{i+j}=\sum_{i=0}^\infty\sum_{j=0}^\infty x^{i+j}$$

And obviously, the coefficient for $x^n$ would be all positive integral solutions: $i+j=n$

And that would be $i=0,1,2,\cdots ,n$.

Therefore we can conclude that the infinite series we're trying to evaluate is $\sum_{n=0}^\infty (n+1)x^n$.

I encountered this calculation in Polya's Problems and Theorems in Analysis, without a proof. However, I think my evaluation is somehow unrigorous. Any more ideas?