I am trying to prove the following version of the Kronecker's Theorem:
Suppose $k$ is a positive integer and $\{1, \theta_0, \dots, \theta_{k-1}\}$ is linearly independent over $\mathbb Q$. Then $\{(n\theta_0+\mathbb Z, \dots, n\theta_{k-1} + \mathbb Z): n \in \mathbb Z\}$ is dense in $T^k$ (where $T$ is the group $\mathbb R /\mathbb Z$ with the natural quotient topology).
I am trying to prove this by induction. I have already proven the case $k=1$ but I don't know what to do in the case $k+1$.