Limit of a periodic function

Let $x\in \R$ be an arbitrary real number, and consider the sequence $f(x),f(x+T),f(x+2T),\ldots$ . By assumption, since $x+n T \underset{n \rightarrow+\infty}{\longrightarrow}+\infty$, it converges to a limit $l\in \R$. But this sequence is constant by periodicity of $f$. This means that $f(x)=l$. Applying the same reasoning for every $x\in \R$, we get that $f$ is constant, equal to $l$.