Proving that $L^p \subset L^q$ when $1 \le q \le p$ [duplicate]

Let $(E,\mathcal{F},\mu)$ be a measure space such that $\mu(E)=1$ and let $L^p=L^p(E, \mathcal{F},\mu)$. Prove that $L^p \subset L^q\text{ if } 1 \le q \le p$.

I let $f \in L^p$. Then $(\int_E |f|^pd\mu)^{1/p} < \infty$. To prove that $f \in L^q$, I should prove that $(\int_E |f|^q d\mu)^{1/q} < \infty $ but I'm unable to do that. I read somewhere that this proof can be done using Holder's inequality but I couldn't do it.

Since $1 \le q \le p < \infty$, $\varphi(x) = x^{p/q}$ is a convex function. Apply Jensen's inequality to get:

$$ \varphi\left(\int \left|f\right|^q \,d\mu\right) \le \int \varphi\left(\left|f\right|^q\right) \,d\mu $$

Hence: $$ \left(\int |f|^q \,d\mu\right)^{p/q} \le \int \left|f\right|^p \,d\mu $$

It follows that $\lVert f \rVert_q \le \lVert f \rVert_p$ and $L^p \subset L^q$.

Consider applying Holder's Inequality to $$\int_E |f|^q \cdot 1.$$ You'll have to be a little clever about the choice of conjugate exponents. As a hint: you're right that you'll want a term of $\int |f|^p$. Can you think of an exponent that will give you that?