MHB Show Inclusion of Measures: Hölder's Inequality

[mathmari]

Hey!

Let $1 \leq p_1 \leq p_2 \leq +\infty$. Show that in a Lebesgue measurable $E\subset R^d$ with $0<m(E)<+\infty$ we have that $L^{p_2} \subsetneq L^{p_1}$.

Using Hölder's inequality I got that $||f||_{p_1} \leq ||f||_{p_2} \mu (E)^{1/p_1 \cdot q}$.

Is this correct so far?? How could I continue to show that $||f||_{p_1} < ||f||_{p_2}$ ?? (Wondering)

Or is there an other way to show this?? (Wondering)

 

[mathmari]

How could I continue to show that $||f||_{p_1} < ||f||_{p_2}$ ?? (Wondering)

Or isn't this that we want to show so that $L^{p_2} \subsetneq L^{p_1}$ ?? (Wondering)