> 
> Let $h:[1, \infty)\rightarrow \mathbb R$ a continuous non-negative
>  function, such that $\int\_{1}^{\infty} h(x)\ dx$ converges.
>  does $h$ must be bounded in $[1, \infty)$?
> 
> 
> 

I tried to prove it by showing that if $\int\_{1}^{\infty} h(x)\ dx$ converges, then by the definition the $\lim\_{b \to \infty}\int\_{1}^{b} h(x)\ dx$ exists. Can I conclude that from the existence of this limit, the function $h$ is bounded?