Let $f:[0,\infty) \to[0, \infty)$ be a continuous function such that $\int_0^{\infty} f(t) dt <\infty$ then find the correct option

Question

Let $f:[0,\infty) \to[0, \infty)$ be a continuous function such that $$\int_0^{\infty} f(t) dt <\infty$$ then which of following are true

(1) the sequence $\{f(n)\}$ is bounded

(2) $f(n) \to0$ as $n \to \infty$

(3) the series $\sum f(n)$ is convergent

i think option 1 and 2 is true, and option 3 is false.but not able to prove 1 and 2 and disprove 3.

any hint please