Can some help me prove or disprove the following assertion which I encountered in research? Thanks!


Let $f:\mathbb R\to\mathbb R$ be an analytic function. If for $\forall c > 0$, we can find some $t'>0$ such that 


$$\int\_{t'}^{t' + 1} {{f^2}(\tau )d\tau } \le c $$
then  

$$\mathop {\lim }\limits\_{t \to \infty } f(t) = 0.$$



A conjecture regarding the integral of the square of an entire function
It's false. Take for example 
$$ 
f(x) =\sum\_{n\in {\Bbb Z}} e^{-n(x-n)^2}. 
$$ 
Clearly $f(n) \ge 1$ for all integers $n$. Since in intervals of length $1$ the function $f$ is large only in a small neighborhood of an integer, it is easy to see that $\int\_x^{x+1} f(t)^2 dt$ tends to zero as $|x| \to \infty$. 


