A lower bound for an exponential series
Let $\gamma >0$ and $r=(r_n)_{n \in \mathbb{Z} } \in \ell^\infty(\mathbb{Z})$ (a bounded and realvalued sequence), where $r_n \geq 0 \ \forall \ n \in \mathbb{Z}$. Consider the function $f(x)=\sum\limits_{n \in \mathbb{Z}} r_n \exp \left (2 \gamma \left (x \frac{n}{2} \right)^2\right)$ for $x \in \mathbb{R}$.
Prove that there exists a constant $C>0$ such that $\sup\limits_{x \in \mathbb{R} }f(x) \geq C \cdot r_\infty$.

The statement doesn't make much sense. You are taking the supremum is the left had side over all x \in R and they say for all x \in R. Did you mean having just f(x) without supremum in the left hand side of what you wish to prove?

yes my bad, I corrected it. thanks

I still think "sup" should be replaced by "inf" for the problem to make sense.

Proving the statement with "sup" is trivially easy.

For inf it is equally trivial.

would it be equally trivial if I didn't require each r_n to be nonnegative?
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