Royden's proof that $L^{p_2}\subseteq L^{p_1}$ if $p_1<p_2$

We are applying Holder's inequality with the conjugate exponents $p:=\frac{p_2}{p_1}$ and $q$. YOu seem to think we're applying it to $p_2$ and $q$ (which is wrong since they aren't even conjugate exponents). Since $1<p,q<\infty$, all the norms are defined using integrals, so: \begin{align} \int_E|f|^{p_1}\cdot 1&\leq \||f|^{p_1}\|_p \cdot \|1\|_q \tag{Holder's inequality}\\ &=\left(\int_E(|f|^{p_1})^{p}\right)^{\frac{1}{p}} \cdot [m(E)]^{1/q}\\ &=\left(\int_E|f|^{p_2}\right)^{\frac{p_1}{p_2}}\cdot [m(E)]^{1/q}\\ &=\left[\left(\int_E|f|^{p_2}\right)^{\frac{1}{p_2}}\right]^{p_1}\cdot [m(E)]^{1/q}\\ &=\|f\|_{p_2}^{p_1}\cdot [m(E)]^{1/q} \end{align} These intermediate steps were "obvious enough" to be left as details to be verified by the reader, and not explicitly mentioned in the form of a theorem/lemma, because really, it's just a matter of checking if one understands the definition of $L^p$ norms.