Constant as periodic

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a periodic function such that $\lim \limits_{x \rightarrow +\infty} f(x) =\ell \in \mathbb{R}$. Prove that $f$ is constant.

Solution

Let us call $T$ the least positive period of $f$. Then it shall hold that:

$$f\left ( x+nT \right )=f(x) \quad , \quad \text{forall } \; x \in \mathbb{R} \;\; \text{and forall} \; n \in \mathbb{N}^*$$

Then it holds that:

\begin{align*}
\lim_{n \rightarrow +\infty} f\left ( x+nT \right )=\lim_{n \rightarrow +\infty} f(x) &\Rightarrow \lim_{n \rightarrow +\infty} f\left ( x+nT \right ) =f(x) \\
 &\!\!\!\!\!\overset{y=x+nT}{=\! =\! =\! =\! \Rightarrow} \lim_{y \rightarrow +\infty} f(y)= f(x) \\
 &\Rightarrow f(x)=\ell
\end{align*}

Hence $f$ is constant.