How to show that this filter is a bandpass filter
You can rewrite the transfer function as
$$ H(\omega) = 2je^{-j\omega/2}\sin(\omega/2) $$
which gives
$$|H(\omega)| = 2|\sin(\omega/2)| $$
I assume we're talking about a discrete-time system, so there is no frequency \$\omega=\infty\$.
We're just considering the range \$\omega\in [0,\pi]\$.
This means that the system is a highpass filter. It's magnitude response increases from $$|H(0)|=0$$ to $$|H(\pi)|=2.$$
Also note that \$H(0)\$ and \$H(\pi)\$ can be easily computed directly from the filter coefficients:
$$H(0) = \sum_{k=0}^{n-1}h_k = 1 - 1 = 0$$
$$H(\pi) = \sum_{k=0}^{n-1}(-1)^kh_k = 1 - (-1) = 2$$