If a function is convergent and periodic, then it is the constant function. I have to prove that if a function f is convergent : $$ \lim\limits_{x\to +\infty} f(x) \in \mathbb{R}$$ and f is a periodic function :
$$\exists T \in \mathbb{R}^{*+}, \forall x \in \mathbb{R}, f(x + T) = f(x)$$
Then f is a constant function.
Actually, this is obvious, but I can't figure out how to prove it formally ..
(I previously thought of this : The link between the monotony of a function, and its limit, but it doen't work because I assume the periodicity hypothesis, and then I use it to prove a contradiction ..)
Thanks !

Let $x\in [0,T)$.
Then 
$$
f(x) = f(x + nT) = \lim_{m\to\infty} f(x + mT)
=  \lim_{X\to\infty} f(X)
$$does not depend on $x$.
