Sequence of r.v {Xn} s.t. every Xi has uniform distribution on (-1,1)
$$Y_{n} = X_{n}/n$$
show that Yn converges in distribution
so far I have $$\mathbb{P}(Y_{n}\leq y)=\mathbb{P}(X_{n}\leq ny)= (ny+1)/2$$ This tends to 0 as n tends to infinity
Is this correct and if so what is the random variable that Yn converges to?
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 $y<0$ $$ \lim_{n\to \infty} \Pr\{Y\le y\}=\lim_{n\to \infty} \Pr\{X\le ny\}=\lim_{n\to \infty} \Pr\{X\le -1\}=0. $$