Let $f_{n}(x)=\frac{n\sqrt{x} \sin{(x)}}{1+nx^{2}}$ show that for all $p \in [1,\infty[$ and for all $n \in \mathbb N$ that $f_{n} \in \mathcal{L}^{p}([0,1])$
My idea:
Let $ n \in \mathbb N$ and $p \in [1,\infty[$
$\vert f_{n}(x)\vert ^{p}\chi_{[0,1]}=\vert\frac{n\sqrt{x} \sin{(x)}}{1+nx^{2}}\vert ^{p}\chi_{[0,1]}=\vert\frac{\sqrt{x} \sin{(x)}}{\frac{1}{n}+x^{2}}\vert ^{p}\chi_{[0,1]}\leq \frac{x^{\frac{p}{2}} x^{p}}{(\frac{1}{n}+x^{2})^{p}}\chi_{[0,1]}\leq \frac{x^{\frac{p}{2}} x^{p}}{(\frac{1}{n})^{p}}\chi_{[0,1]}=n^{p}x^{\frac{3}{2}p}\chi_{[0,1]}\in \mathcal{L}^{1}$
My question:
Have I indeed show that $f_{n} \in \mathcal{L}^{p}([0,1])$ since my dominating integrable function still contains $n$? I would argue it does not because $n$ is a constant, but I am also unsure as $n \to \infty$?
Any reassurances, correction are greatly appreciated.
The question doesn't ask you to find out what happens as $n \to \infty$. It asks you to prove that $|f_n|^{p}$ is integrable for each fixed $n$ and each fixed $p$. What you have done is correct but you can make it a little simpler. Note that $|f_n(x)| \leq n$ for all $n$ for all $x$. [Use: $1+nx^{2} \geq 1, \sqrt x \leq 1$ and $|\sin\, x| \leq 1$]. This is enough to draw the conclusion.