It is given that the measure $ \mu(X) \lt \infty$, and $1 \le p \lt q \le \infty$. The problem has two parts, whereas in the first part where we need to show that $ \mathscr{L}^q \subseteq \mathscr{L}^p $ is clear to me.
It was given that a function sequence $ f_n \rightarrow f$ in $\mathscr{L}^q $, we are supposed to show that $ f_n \rightarrow f$ in $\mathscr{L}^p $, where I assume that I should use the subset property, but I am not very clear with the method of starting the proof itself.