convergence in distribution of transformation of random variable
Your approach has a flaw. You have used $\Pr\{X\le u\}=\frac{u+1}{2}$ for all $u\in \Bbb R$, while this only holds for $u\in [-1,1]$. For this reason, for $y>0$ you should write $$ \lim_{n\to \infty} \Pr\{Y\le y\}=\lim_{n\to \infty} \Pr\{X\le ny\}=\lim_{n\to \infty} \Pr\{X\le 1\}=1 $$ and for $y<0$ $$ \lim_{n\to \infty} \Pr\{Y\le y\}=\lim_{n\to \infty} \Pr\{X\le ny\}=\lim_{n\to \infty} \Pr\{X\le -1\}=0. $$