Pointwise convergence of a sequence of trigonometric polynomials with bounded number of nonzero terms

Question

Let $K$ be a fixed integer, and $\mathcal{F}$ the set of trigonometric polynomials with at most $K$ nonzero terms. Let $(f_n)$ be a sequence in $\mathcal{F}$ converging pointwise (on $\mathbb{R}$) to some continuous function $f$. Then there is a dichotomy

• if $f \in \mathcal{F}$ then $|f_n - f|_{L^{\infty}} \longrightarrow 0$ as $n \rightarrow + \infty$.

• if $f \notin \mathcal{F}$ then $|f_n - f|_{L^{\infty}} \longrightarrow + \infty$ as $n \rightarrow + \infty$.

Question : Can the second case actually happens ? I've not been able to provide any example so far...