$\textbf{6.Q.}$ Let $f_n \in L_p(X, \textbf{X}, \mu)$, $1 \leq p < \infty$ and let $\beta_n$ be defined for $E \in \mathbf{X}$ by
$$\beta_n(E) = \left\{ \int_E |f_n|^p d\mu \right\}^{\frac{1}{p}}$$
Show that $|\beta_n(E) - \beta_m(E)| \leq ||f_n - f_m||_p$. Hence, if $(f_n)$ is a Cauchy sequence in $L_p$, then $\lim \beta_n(E)$ exists for each $E \in \mathbf{X}$.
This is what I thought:
By Minkowski's inequality, I ensure that
$$\left| ||f_n||_p - ||f_m||_p \right| \leq ||f_n - f_m||_p$$
It seems that $|\beta_n(E) - \beta_m(E)| \leq \left| ||f_n||_p - ||f_m||_p \right|$ it's true, but I can't prove this. Is this the way to prove the inequality of this exercise? I would like to a hint.
Note that by Minkowski's inequality we have $$\left| ||g_n||_p - ||g_m||_p \right| \leq ||g_n - g_m||_p$$ where $$g_n:=\chi_E f_n$$
Therefore,by definition we have $$|\beta_n(E) - \beta_m(E)|=\left| ||g_n||_p - ||g_m||_p \right| \leq ||g_n - g_m||_p$$
But $$||g_n - g_m||_p=(\int_X |g_n-g_m|^p d\mu)^\frac{1}{p}=(\int_X|f_n-f_m|^p\chi_E\ d\mu)^\frac{1}{p}≤(\int_X|f_n-f_m|^p\ d\mu)^\frac{1}{p}=||f_n-f_m||_p$$ Since $|f_n-f_m|\chi_E≤|f_n-f_m|$.