Need guidance on this problem. Given the following ordinals, determine the order relation (and equalities) - ($\Omega = 2^{\aleph_0}$)
$\omega_1$, $\Omega \cdot 3$, $\omega \cdot \omega_1$, $3^{\Omega}$, $\Omega^{\omega_1}$
My solution (partial):
$\Omega\cdot 3 \gt \Omega\cdot 2 \gt \Omega$, $3^{\Omega} = \lim_{n \lt \Omega}3^{n} = \Omega$ ($\Omega$ is a limit ordinal)
I'm not certain about how to deal with $\omega \cdot \omega_1$ and $\Omega^{\omega_1}$
The inequalities:
$\Omega\cdot 3 \gt 3^{\Omega} \gt \omega_1$
Thanks!