Show that the sequence $(\frac {\sin nx}{ \sqrt{n}})$ is uniformly convergent on $\mathbb R$.
$|\sin nx / \sqrt{n}| \leq |1/ \sqrt{n}|$ $\forall x \in \mathbb R$ since $|\sin nx| \leq 1$.
Let us choose $n>1/\epsilon^2 > 0$ $\implies$ $\epsilon > 1/\sqrt{n}$.
Then for a given $\epsilon > 0$,
$|\sin nx / \sqrt{n}| \leq |1/ \sqrt{n}| < \epsilon$ $\forall x \in \mathbb R$.
Hence, it is uniformly convergent to $0$ $\forall x \in \mathbb R$.
I don't know whether I am right or wrong, please suggest.