Suppose that $X_1,X_2,\cdots,X_n$ are iid sequence with pdf $\frac{2}{\pi (1+x^2)}\cdot 1_{(0,+\infty)}(x)$. Denote $S_n$ as $S_n:=X_1+X_2+\cdots+X_n$. Prove that there exits $c>0$ such that $$\frac{S_n}{n\log n}\rightarrow c\quad \text{in probability}$$ and calculate $c$.
My solution: Choose another iid sequence $Y_1,Y_2,\cdots,Y_n$ such that $X_n,Y_n$ are independent and have same distribution. Let $X_n^s:=X_n-Y_n$. I have proved that $$\frac{X_1^s+X_2^s+\cdots+X_n^s}{n\log n}\rightarrow 0\quad \text{in probability}.$$ Therefore, $$\frac{S_n}{n\log n}-m_n\rightarrow 0\quad \text{in probability}$$ where $m_n$ is the medin of $\frac{S_n}{n\log n}$. I want to show $m_n\rightarrow c$ but I can't.