$L^q(X) \subset L^p(X)$ if $p\leq q$ and $\mu(X) < \infty$. Let $(X,\mathfrak{M},\mu)$ be a measure space. Show that if $\mu(X) < \infty$ then $L^q(X) \subset L^p(X)$ for $1\le p \le q \le \infty$.  
Is it enough to define $$E_0 = \{x \in X \, : \, 0 \leq |f(x)| < 1\}$$ and
$E_1 = X \setminus E_0$ so that $\mu(E_0), \, \mu(E_1) < \infty$ and $E_0$ is measurable as $E_0 = \{x \in X : |f(x)| \geq 0 \} \cap \{x \in X : |f(x)| \leq 1 \}$ and $f$ is measurable. Therefore if $f \in L^q$,
\begin{eqnarray*}
||f||^p_{L^p} &=& \int_{E_0}|f|^p \, d\mu + \int_{E_1} |f|^p \, d\mu \\
&\leq& \int_{E_0}|f|^p \, d\mu + \int_{E_1} |f|^q \, d\mu \\
&<& \mu(E_0) \,\, + ||f||_{L^q}^q \\
&<& \infty.
\end{eqnarray*}
which implies $f \in L^p$.  
Doe this suffice? I've seen a proof using Holder's inequality (that has now been given as an answer for those interested), that is much shorter, but does this suffice?

Your proof is fine. You can also try to show that if $p<q$ then  $$\lVert f\rVert_p \leqslant \lVert f\rVert_q \mu(X)^{1/p-1/q}$$
by using Hölder's inequality.  This gives a bit more information than your proof. In particular for $X$ a probability space gives $p\mapsto \lVert f\rVert_p$ is increasing.  
