A lower bound

Let $\gamma >0$ and $r=(r_n)_{n \in \mathbb{Z} } \in \ell^\infty(\mathbb{Z})$ (a bounded and real-valued sequence). Consider a function $f$ such that $f(x)=\sum\limits_{n \in \mathbb{Z}} r_n \exp \left (-2 \gamma \left (x- \frac{n}{2} \right)^2\right)$ for $x \in \mathbb{R}$.

 
Prove that there exists a constant $C>0$ such that $\sup\limits_{x \in \mathbb{R} }|f(x)| \geq C \cdot ||r||_\infty$ for all $x\in \mathbb{R}$.

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  • sorry for the late reply. I made a mistake, the sum is supposed to be equal to the absolute value of f. I posted another question with the corrected version.

The answer is accepted.