Let $f_{n}(x)$, for $n \geq 1$, be a sequence of continuous non negative functions on $[0, 1]$ such that
$\displaystyle \lim_{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) \text{d}x$
Which of the following statements is always correct?
- $f_{n} \rightarrow 0$ uniformly on $[0, 1]$
- $f_{n}$ may not converge uniformly but converges to $0$ point-wise
- $f_{n}$ will converge point-wise and the limit may be non-zero
- $f_{n}$ is not guaranteed to have a point-wise limit
