$L_p$ Norms and Holder's Inequality question

Question

Suppose that $-∞ < a < b < ∞$ and $1 < p < q < ∞$. Let $$L_p[a,b] = \{ f :\Bbb R \to \Bbb R : \left( \int_a^b\left|f\left(x\right)\right|^p~\mathrm dx\right)^{\frac{1}{p}} < ∞ \}.$$

Show that $L_q ⊂ L_p$.

Any ideas? I'm not sure what the steps are to show that something is $⊂$ something else.

Answer 1

Let $f \in L^q$.

If $|f(x)| \le 1$ then $|f(x)|^p \le 1$

If $1 \le |f(x)| $ then $1 \le |f(x)|^p \le | f(x) |^q$

Whatever $|f(x) |$ is, $|f(x) |^p \le 1 + | f(x) |^q$

Use this fact.