Sample correlation coefficient of i.i.d Cauchy variables

Question

Let $X_1,X_2, \dots, X_n, \dots$ be i.i.d Cauchy random variables, and let $Y_1,\dots, Y_n,\dots$ be i.i.d Cauchy random variables independent of the $X_i$'s.

Let $\rho_n$ be the sample correlation between the $X$'s and $Y$'s based on the first $n$ samples. [changed again, there was a bug in my code] Simulation studies show that $\rho_n$ converges to $0$. The image below shows shows the values of $\rho_1,\dots,\rho_{10000}$ for one sample of $X_1,\dots,X_{10000}$ and $Y_1,\dots,Y_{10000}$.

This suggests $P(\lim_{n\to\infty} \rho_n = 0) = 1.$ I am looking for a proof of the above claim.

/*---code to produce the plot above---*/
proc iml;
n  = 10000;
call randseed(1242525);
x = j(1,2);
d = {};
rho = {};
do i = 1 to n;
call randgen(x,"cauchy");
d = d // x;
rho = rho // (corr(d)[2]);
end;

rho = t(1:n) || rho;

create rho from rho[colname={i rho}];
append from rho;
close;
quit;

title "Trajectory of Sample Correlation";
proc sgplot data=rho;
series x=i y=rho;
run;