We consider the space $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { d } \right)$ of functions $f : \mathbb { R } ^ { d } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of their variables, equipped with the uniform norm $\| f \| _ { \infty } = \sup _ { \theta \in \mathbb { R } ^ { d } } | f ( \theta ) |$. A trigonometric polynomial (in $d$ variables) is any function of the form $\theta \mapsto \sum _ { k \in K } c _ { k } e ^ { 2 i \pi k \cdot \theta }$ where $K$ is a finite subset of $\mathbb { Z } ^ { d }$. We work in dimension $d = 2$. The subspace $\mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$ of $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ consists of functions of the form $\theta = \left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \sum _ { i = 1 } ^ { n } f _ { i } \left( \theta _ { 1 } \right) g _ { i } \left( \theta _ { 2 } \right)$, where $n \in \mathbb { N } ^ { * }$ and $f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \in \mathscr { C } _ { \text {per} } ( \mathbb { R } )$. We admit that trigonometric polynomials in one variable are dense in $\mathscr{C}_{\text{per}}(\mathbb{R})$.
Show that the set of trigonometric polynomials in two variables is dense in $\mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$.