How can I show that, for $p \in [1,2], f \in L^p$ can be expressed as $f = f_1 + f_2$, where $f_1 \in L^1, f_2 \in L^2$?
My idea was that $f \in L^p \implies \int|f(x)|^p dx < \infty$. Then, as $p \geq 1$, whenever $x $ is such that $|f(x)| \geq 1$, then $|f(x)|^p \geq |f(x)|$, so that by monotonocity, $\int |f(x)|1_{|f(x)| \geq 1} < \int |f(x)^p|1_{|f(x)| \geq 1} < \infty$.
Similarly, since $p \leq 2$, then for $x$ such that $|f(x)| \leq 1$, we have $|f(x)|^p \geq |f(x)|^2$, so that $\int |f(x)|^21_{|f(x)| \leq 1} < \int |f(x)^p|1_{|f(x)| \leq 1} < \infty$.
I have two questions now:
How does this imply $f \in L^2_{loc}$?
I think this argument works for any $p \geq1$, if we make a slight change and take $f^2 \in L^{p_2}, p_2 \geq p$ instead. Am I missing something in the proof, or is this general result indeed valid?