It holds that $C(\mathbb{R})$ is a Banach space, or not? Then $f_n(x)\in X\subseteq C(\mathbb{R})$. So a Cauchy sequence $\{f_n\}$ of $C(\mathbb{R})$ converges to $f\in C(\mathbb{R})$.
It is left to show that if $f_n$ is $T$-periodic then $f$ is $T$-periodic.
Suppose that $f$ is not $T$-periodic, then $f(t)\neq f(t+T)$ for $t\in \mathbb{R}$.
Since $f_n$ is $T$ periodic, we have that $f_n(t)=f_n(t+T)$. We take the limit $n\rightarrow\infty$ and get $f(t)=f(t+T)$, a contradiction.
That means that a Cauchy sequence $\{f_n\}$ of $X$ converges to $X$, and so $X$ is complete.