$\mathbb{C} = \bigcup_{\omega \in L } (\omega + \mathcal{F})$

Question

Let $\mathcal{F}$ be a fundamental parallelogramm of the lattice $L$. Show \begin{align*} \mathbb{C} = \bigcup_{\omega \in L } (\omega + \mathcal{F}) \, . \end{align*}

In the solutions they wrote The claim follows directly from the well-known fact that for any real $n$ $x$ there exists an integer $n$ with $0 ≤ x − n ≤ 1.$

How can one write the proof formally?

The definition of the fundamental parallelogramm is \begin{align*} \mathcal{F} = \mathcal{F}(\omega_1,\omega_2) = \{ t_1 \omega_1 + t_2 \omega_2 ; 0 \leq t_1,t_2 \leq 1 \} \end{align*}

Answer 1

Proof. Let $z \in \mathbb{C}$ then : \begin{align*} z= s_1 \omega_1 + s_2 \omega_2 , \quad s_1, s_2 \in [0,1) \end{align*} For any real number $s_1,s_2$ there exists $n_1,n_2$ so that \begin{align*} 0\leq s_1 - n_1 \leq 1, \quad 0 \leq s_2 - n_2 \leq 1 \end{align*} and then we can write \begin{align*} \underbrace{s_1 \omega_1 + s_2 \omega_2}_{=z\in \mathbb{C}} = (s_1 -n_1 ) \omega_1 + (s_2-n_2) \omega_2 \end{align*} Is that correct?

Answer 2

Every complex number can be written in the form $$ z = t_{1}\omega_{1} + t_{2}\omega_{2} $$ for some real $t_{1}$ and $t_{2}$. Decompose these (uniquely) as in your hint, into an integer plus a real number in $[0, 1)$.