Let $f : [0,\infty) \to [0,\infty)$ be submultiplicative, i.e. $f(s+t) \leq f(s)f(t)$ for all $t,s\geq 0.$ Further assume $\lim_{t \to \infty} f(t)=0.$ Then show that $f$ decay exponentially.
My attempt: We want to show there exists $r>0,M \geq 1$ such that $f(t)\leq M e^{-rt}$ for all $t \geq 0.$ If not, then for each $n \in \mathbb{N},$ there exists $t_n\geq 0$ such that $$f(t_n)>e^{-nt_n}.$$ Submultiplicativity of $f$ implies $$\left[f\left(\frac{t_n}{n}\right)\right]^n\geq f(t_n)>e^{-nt_n}$$ and hence $$f\left(\frac{t_n}{n}\right)>e^{-t_n}.$$
I'm not sure how to proceed from here.