Limit of an almost periodic function

Consider the function: $$ f(t)=\sum_{\nu\in F} c_\nu e^{i \nu t} $$ where $F\subset \mathbb{R}$ is a countably finite or infinite set (but $F\neq\{0\}$) and $c_\nu\in \mathbb{C}$.

The question is to give a simple argument which shows that the limit $\lim_{t\to\infty}f(t)$ does not exist. A way to proceed would be to prove that $f(t)$ is almost-periodic: $$ |f(t)-f(t+\tau_\epsilon)|<\epsilon $$ for any $\epsilon$ and $t$. However I do not see a simple way to obtain this latter result avoiding some theorems and lemmas. Any ideas?

Solution 1:

Let $$f_N(t)= \sum_{n=1}^N b_n e^{i w_n t}$$ We find that $$ \begin{eqnarray}\int_{-1}^1 f_N(kt)e^{-i w_1k t}dt&=& \sum_{n=1}^N b_n \int_{-1}^1 e^{i w_nkt}e^{-i w_1k t}dt \\ &=& \sum_{n=1}^N b_n 2 \frac{\sin( (w_n-w_1)k)}{(w_n-w_1)k} \end{eqnarray}$$ As $k\to \infty$ the RHS $\to 2 b_1$, which implies that $$\lim\sup_{t\to \infty}|f_N(t)|\ge |b_1|$$

If your series $f(t)=\sum_{\nu\in F} c_\nu e^{i \nu t}$ converges uniformly then for any $a$ $$\lim \sup_{t\to \infty} |f(t)-a| \ge \sup_{\nu\ne 0} |c_\nu|$$