Function in $\mathcal{L}^p(\mu)$ for all $1\leq p \leq 2$

Question

Let (X, $\mathscr{A}$, $\mu$) be a measure space and f $\in$ $\mathcal{L}^1(\mu)\cap \mathcal{L}^2(\mu)$.

Can I ask how to show that $f \in \mathcal{L}^p(\mu)$ for all $1\leq p\leq 2$?

Answer 1

$$|f(x)|^p \leqslant |f(x)| \mathbb 1_{\{|f(x)|\leqslant 1\}}+|f(x)|^2 \mathbb 1_{\{|f(x)|\gt 1\}}.$$