$L^p$ convergence follows from $L^2$ convergence plus boundedness of $L^\infty$?

Question

Let $f_n: \mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions with $(f_n)\subset L^2\cap L^\infty$. Further let there be an $f \in L^2( \mathbb{R})$ and a constant $C>0$ with the characteristics $f_n\rightarrow f$ in $L^2( \mathbb{R})$ and $\|f_n\|_{L^\infty}\leq C$.

A) Show that $f\in L^\infty( \mathbb{R})$ B) Show that $f_n\rightarrow f$ in $L^p( \mathbb{R})$ for $2\leq p <\infty$

Answer 1

A) I claim $|f| \le C$ a.e. Otherwise, there is a set of positive measure on which $|f| > k > C$ for some $k$, implying a lower bound on $\|f - f_n\|_2^2$.

B) Note that for any $x \in [0,2C]$, $x^p \le (2C)^{p-2} x^2$. Then

$$\|f_n - f\|_p^p \le (2C)^{p-2} \|f_n - f\|_2^2$$