Limit of integral from t/2 to t

Question

So I've been trying to understand some calculus, and I found this. Supposedly if $\int_{0}^{\infty}f(x)<\infty$ and $f(x)\geq0$ then we have $$\lim_{t \to \infty}\int_{\frac{t}{2}}^{t} f(x)\,dx=0$$ Is that true? And if so, why is that true?

Answer 1

This is true if we assume the that the integral of f converge

Hint :

Cauchy criterion

Answer 2

Hint: $$\int_{0}^{t} f(x)\,dx=\int_0^{\frac{t}{2}} f(x)\,dx+\int_{\frac{t}{2}}^{t} f(x)\,dx$$