Suppose $f:\mathbb R\to \mathbb C$ is a Schwartz function such that there is $C>0$ such that all of its Schwartz seminorms are bounded by $C$: $$ \sup_{m,n}\sup_x |x|^n |f^{(m)}(x)|\leq C. $$ Must $f$ be 0?
My intuition is true, and I think it may be proved using complex analysis.