Convergence of sequence of functions with supremum metric

Question

Let $f_n(x)=\sin(nx), x \in [0, \pi]$. Does this sequence converge with the supremum metric $d(f,g)=\sup_{x \in [0, \pi]}|f(x)-g(x)|$? I think there is a result stating that convergence in supremum metric is equivalent to uniform convergence with standard metric, so by this we know that $\sin(nx)$ does not even converge pointwise in $[0, \pi]$, so it does not converge uniformly and therefore does not converge with the supremum norm, is that correct?