If the limit of the subtraction of two periodic function is ZERO when x goes to INFINITY, is these two functions equals to each other?

Question

If $f:\Bbb R\to \Bbb R$ and $g:\Bbb R\to \Bbb R$ are periodic functions such that $$\lim_{x \to + \infty }(f-g)(x)=0$$ Prove $f=g$, or give a counter example.

By the way, please notice that the period of $f$ and $g$ may not be equal nor rational.

Answer 1

The result is right. Here follows the proof: Let $T_1,T_2$ be the period of $f$ and $g$. Then for $\forall\ x\in\mathbb{R}$, $$\lim_{n\to\infty}\left(f(x+nT_1)-g(x+nT_1)\right)=0,$$ so $f(x)=\lim\limits_{n\to\infty}g(x+nT_1);$ $$\lim_{n\to\infty}\left(f(x+nT_2)-g(x+nT_2)\right)=0,$$ so $g(x)=\lim\limits_{n\to\infty}f(x+nT_2).$ Thus$$f(x)-g(x)=\lim_{n\to\infty}\left(g(x+nT_1)-f(x+nT_2)\right)$$ $$=\lim_{n\to\infty}\left(g(x+nT_1+nT_2)-f(x+nT_1+nT_2)\right)=0.$$ So we get $$f=g.$$