Frequency range for set of all discrete-time sinusoids
assuming $\omega$ is real, if the form of $x[n]$ is how you've defined it above, your thinking is correct.  doesn't matter if it's $+\omega$ or $-\omega$, the sinusoid behaves exactly the same.  if instead, the sinusoid had form

$$ x[n] = e^{j \omega n}  \ ,$$ 

then your teacher would be correct.