Let $\omega_1, \omega_2$ be two nonzero complex numbers. If $\omega_1/\omega_2$ is not real, prove that there exists a constant $k >0$ such that for all $n,m \in \mathbb Z$ $|m\omega_1+n\omega_2| \geq k(|m|+|n|)$.
This problem arises in proving that the Weierstrass p function $\wp$ converges absolutely and uniformly on any compact set. We may assume without loss of generality that $\omega_1 =1$ and $\omega_2$ not real, by replacing $\omega_1$ and $\omega_2$ and $\omega_1 /|\omega_1|$ and $\omega_2 /|\omega_2|$ respectively if necessary. Also, it suffices to prove the inequality for all sufficiently large $n$ and $m$. I looked at Stein and Shakarchi's Complex Analysis and I didn't get their method of proving.