Let $f\in C_c^{0}(\mathbb{R}^n)$ , and $(x_n)$ be a sequence such that $||x_n|| \to \infty$ is true that $f(x_n) \to 0$ when $n \to \infty$

Question

Let $f$ be a compact support continuous function, and $(x_n)_{n \in \mathbb{N}}$ be a sequence such that $||x_n|| \to \infty$ it is true that $f(x_n) \to 0$ when $n \to \infty$ ?

I think to these fact it is true because like the sequence is unbounded and the support of $f$ is bounded (because is compact) for $N$ large eventually the sequence $x_n$ stay out of the support of $f$ and hence for $n \geq N$ the sequence $f(x_n)$ is zero These is correct?