This is Problem 3.4.13 from $\textbf{ Problems in Real Analysis Advanced Calculus on the Real Axis.}$
Let $f_1, f_2,\ldots,f_n:\mathbb{R}\longrightarrow\mathbb{R} $ be periodic functions such that \[ \lim_{x\to\infty} \left(f_1(x)+f_2(x)+\cdots f_n(x)\right) = 0 \] Prove that $f_1=f_2=\cdots=f_n$
However, is it alright to reason as follows?
If $n\leq 2$, then it is obvious. If $n\geq 3$ choose two elements from a Hamel basis $\mathcal{H}$, say $a, b$. Write $x = \sum_{y\in \mathcal{H}} (x,y)\, y$ then \[ f_1(x) = (x,a)a-(x,b)b \] \[ f_2(x) = \sum_{y\neq a} (x,y)\,y \] \[ f_3(x) = -\sum_{y\neq b} (x,y)\,y \] and $f_i(x) = 0$ for all $i\geq 4$
And I am currently at a loss if the problem is wrong. It seems to me that not only the functions are different but they don't share a period.
Edit
A Hamel basis is a basis of $\mathbb{Q}$ vector space $\mathbb{R}$