Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt Characteristic Functions. I do not know anything about Complex Analysis, Residue Theorem, etc.
I would want to prove the statement only using Real Calculus. Feel free to use Double Integrals if you please.
On searching, I found this. However, I was wondering if I could get some help directly on the convolution formula:
Here I have supposed that $X,Y$ are Independent Standard Cauchy. But I think the general formula can be derived easily after some substitutions. I need some help on how to proceed from $(1)$.
EDIT: Just as what the hint in the hyperlink said, I got the answer using that hint. However, I am not quite sure that the hint is algebraically correct. Maybe there has been some typing mistake in the book.