if $f(x)\geq 0$ is uniformly continuous on $[1, \infty)$ and $\int_1^{\infty}f(x)dx$ converges, then $\lim_{x\rightarrow\infty}f(x)=0$

Question

Prove or disprove: if $f(x)\geq 0$ is uniformly continuous on $[1, \infty)$ and $\int_1^{\infty}f(x)dx$ converges, then $\lim_{x\rightarrow\infty}f(x)=0$

I'm pretty sure it's true because if f is only continuous, all opposite examples I know use infinite amount of x valus where f(x)=1 for example (like a triangle of height 1 with a smaller base for each natural x). But such an example wouldn't be possible because the function would be too "steep" around natural x's.

I'm looking for a hint on how to formalize this, I was thinking about Lagrange but obviously f isn't necessarily differentiable...