About convergence in norm of the Fourier Transform Duoandikoetxea's Fourier Analysis, on page 59 (Corollary 3.7) says that:
\begin{equation}
\lim_{R \rightarrow \infty}\big\|S_{R}\,f - f\big\|_{p} = 0
\end{equation}
for $1<p<\infty$, where $S_{R}\,f \:$ is such that $\widehat{S_{R}\,f}\left(y\right)$ = $\chi_{(-R,R)}\left(y\right) \,\hat{f}(y)$. He also proves that there is a constant $C_{p}$ such that $||\,S_{R}\,f\,||_{p} \le C_{p}\|\,f\|_{p}$, and that this $C_{p}$ doesn't depend on $R$.
After that he says that this is not true when $p=1$, but he doesn't give a counterexample. What function would contradict this when $p=1$? 
Another thing, he obtains the limit above as a consequence of $\|\,S_{R}\,f\,\|_{p} \leq C_{p}\,\|f\|_{p}$. I can prove the limit above for all $f \in L^{p}$ if I know that it is valid for a dense subset, let's say, $S(\mathbb{R})$ (Schwartz class) or $C^{\infty}_{c}$ by approximating $f$ using such functions. How do I prove the limit above for $f$ in any of these spaces?

For the second part, it follows from the fact that you have inversion formula if $f$ is in Schwartz, i.e. 
$$f(x)=\int\hat f(\xi)e^{2\pi ix\xi}d\xi$$
