Statements on function with finite integral over $[0, \inf[$ 
Let $f: [0, \infty[\to[0, \infty[$ be a continuous function such that:
  $$\int_0^\infty f(x) dx < \infty$$
  Which of the following statements are true? 



*

*The sequence $\{f(n)\}_{n\in\mathbb{N}} $ is bounded. 

*$f(n) \to 0$ as $n\to \infty $

*The series $\sum_{n=1}^\infty is convergent.


Intuitively i feel each option is true.
For (a),if $f(n)$ is unbounded,since $f$ is non negative,the integral cannot be finite.
For (b),if $f(n)$ does not tend to $0$,then again integral cannot be finite.
For (c),since the series should take a value lesser than or equal to the integral of $f$,the series should be convergent(since partial sums must be bounded).
However the answer says none of the options are correct! Where am i going wrong?

$a)$ In the words of angryavian (who deleted his/her answer), consider as a counterexample a function which has infinitely many, thin spikes, progressively taller spikes and is zero elsewhere. The height of the spikes is unbounded, but the width at the bottom of each spike could be made small enough to make the area under the $k^{th}$ spike equal to $\dfrac {1}{2^k}$, so that the integral is $\dfrac 12 + \dfrac 14 + \dfrac 18 + \cdots$ which is finite.
$b)$ The same function as in $a$. We see that for that function, the limit doesn't exist.
$c)$ Same function again, just makes the peaks of infinitely many of the spikes on the integers.
