Sequence of measurable $\&$ continuous functions defined on $[0,1]$ Let $\{f_n\}$ be a sequence of measurable $\&$  continuous functions from $[0,1]$ to $[0,1]$. Assume $f_n \rightarrow f$ pointwise. Is it true/false that,


*

*$f$ is Riemann integrable  $\& \int _{[0,1]}f_n \rightarrow \int_{[0,1]}f$?

*$f$ is Lebesgue integrable $\& \int _{[0,1]}f_n \rightarrow \int_{[0,1]}f$?
My work:
For $(1),$ I came up with a counter-example $f_n(x)=nx(1-x^2)^n$ on $[0,1]$ as $f_n \rightarrow 0$ but $\int _{[0,1]}f_n=\frac12$ (but is $f_n(x) \in [0,1]$ for all $x$??)
For $(2),$ I think this holds since,
1.The continuous functions over a closed bounded interval is R.I & hence L.I
2.measure space is finite. 


*$\{f_n\}$ is uniformly bounded and it converges to $f$ pointwise. 


So,by DCT, this holds true.
Am I correct? Also what about my choice of function for case (1)?

$$
\text{Let } f_n(x) = \begin{cases} 2n\left(1-|2nx-1| \right) & \text{if } 0\le x\le1/n, \\[6pt] {} 0 & \text{otherwise.} \end{cases}
$$
Then
$$
\lim_{n\to\infty} \int\limits_{[0,1]} f_n(x)\,dx = 1 \ne 0 = \int\limits_{[0,1]} \lim_{n\to\infty} f_n(x)\,dx.
$$
